---

${{{\mathrm{\Gamma }}^t}_t}_t$
factor out index raise
${g^t}^a{{{\mathrm{\Gamma }}_a}_t}_t$
substitute definition of connection
$\frac{1}{2}{g^t}^a\left(-{{g_t}_t}_{,a}+2{{g_a}_t}_{,t}\right)$
split the index a into t and j
$\frac{1}{2}\left(-{g^t}^j{{g_t}_t}_{,j}+{g^t}^t{{g_t}_t}_{,t}+2{g^t}^j{{g_j}_t}_{,t}\right)$
replace ADM metric definitions
$\frac{1}{2}\left(-\frac{{\beta }_j}{{\alpha }^2}{(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,j}+{(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,t}\frac{-1}{{\alpha }^2}+2\cdot \frac{{\beta }_j}{{\alpha }^2}{{\beta }_j}_{,t}\right)$
raise ${\beta }_j$
$\frac{1}{2}\left(-\frac{{\beta }_j}{{\alpha }^2}{(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,j}+{(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,t}\frac{-1}{{\alpha }^2}+2\cdot \frac{{\beta }_j}{{\alpha }^2}{\left({{\gamma }_j}_k{\beta }^k\right)}_{,t}\right)$
simplify...
${\alpha }_{,t}\frac{1}{\alpha }+-1\cdot \frac{1}{2}{\beta }^l{{\beta }^k}_{,t}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,t}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^k{{\beta }^l}_{,t}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+{\alpha }_{,j}{\beta }_j\frac{1}{\alpha }+{\beta }_j{\beta }^k{{{\gamma }_j}_k}_{,t}\frac{1}{{\alpha }^2}+{\beta }_j{{\beta }^k}_{,t}{{\gamma }_j}_k\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }_j{\beta }^l{{\beta }^k}_{,j}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }_j{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,j}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }_j{\beta }^k{{\beta }^l}_{,j}{{\gamma }_k}_l\frac{1}{{\alpha }^2}$
relabel...
${\alpha }_{,t}\frac{1}{\alpha }+-1\cdot \frac{1}{2}{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,t}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^k{{\beta }^l}_{,t}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^l{{\beta }^k}_{,t}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+{\alpha }_{,j}{\beta }_j\frac{1}{\alpha }+{\beta }_j{\beta }^k{{{\gamma }_j}_k}_{,t}\frac{1}{{\alpha }^2}+{\beta }_j{{\beta }^k}_{,t}{{\gamma }_j}_k\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }_j{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,j}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }_j{\beta }^l{{\beta }^k}_{,j}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }_j{\beta }^k{{\beta }^l}_{,j}{{\gamma }_k}_l\frac{1}{{\alpha }^2}$
for ${\beta }^i=0$
$\frac{1}{\alpha }{\alpha }_{,t}$

---

${{{\mathrm{\Gamma }}^t}_t}_i$
factor out index raise
${g^t}^a{{{\mathrm{\Gamma }}_a}_t}_i$
substitute definition of connection
$\frac{1}{2}{g^t}^a\left({{g_a}_t}_{,i}+{{g_a}_i}_{,t}-{{g_t}_i}_{,a}\right)$
split the index a into t and j
$\frac{1}{2}\left({g^t}^j{{g_j}_t}_{,i}+{g^t}^j{{g_j}_i}_{,t}-{g^t}^j{{g_t}_i}_{,j}+{g^t}^t{{g_t}_t}_{,i}\right)$
replace ADM metric definitions
$\frac{1}{2}\left(\frac{{\beta }_j}{{\alpha }^2}{{\beta }_j}_{,i}+\frac{{\beta }_j}{{\alpha }^2}{{{\gamma }_j}_i}_{,t}-\frac{{\beta }_j}{{\alpha }^2}{{{\gamma }_t}_i}_{,j}+{(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,i}\frac{-1}{{\alpha }^2}\right)$
raise ${\beta }_j$
$\frac{1}{2}\left(\frac{{\beta }_j}{{\alpha }^2}{\left({{\gamma }_j}_k{\beta }^k\right)}_{,i}+\frac{{\beta }_j}{{\alpha }^2}{{{\gamma }_j}_i}_{,t}-\frac{{\beta }_j}{{\alpha }^2}{{{\gamma }_t}_i}_{,j}+{(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,i}\frac{-1}{{\alpha }^2}\right)$
simplify...
$\frac{1}{2}{\beta }_j{{{\gamma }_j}_i}_{,t}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }_j{{{\gamma }_t}_i}_{,j}\frac{1}{{\alpha }^2}+{\alpha }_{,i}\frac{1}{\alpha }+\frac{1}{2}{\beta }_j{\beta }^k{{{\gamma }_j}_k}_{,i}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,i}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^l{{\beta }^k}_{,i}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^k{{\beta }^l}_{,i}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_j{{\beta }^k}_{,i}{{\gamma }_j}_k\frac{1}{{\alpha }^2}$
relabel...
$\frac{1}{2}{\beta }_j{{{\gamma }_j}_i}_{,t}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }_j{{{\gamma }_t}_i}_{,j}\frac{1}{{\alpha }^2}+{\alpha }_{,i}\frac{1}{\alpha }+-1\cdot \frac{1}{2}{\beta }^k{{\beta }^l}_{,i}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_j{\beta }^k{{{\gamma }_j}_k}_{,i}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_j{{\beta }^k}_{,i}{{\gamma }_j}_k\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,i}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^l{{\beta }^k}_{,i}{{\gamma }_k}_l\frac{1}{{\alpha }^2}$
for ${\beta }^i=0$
$\frac{1}{\alpha }{\alpha }_{,i}$

---

${{{\mathrm{\Gamma }}^i}_t}_t$
factor out index raise
${g^i}^a{{{\mathrm{\Gamma }}_a}_t}_t$
substitute definition of connection
$\frac{1}{2}{g^i}^a\left(-{{g_t}_t}_{,a}+2{{g_a}_t}_{,t}\right)$
split the index a into t and j
$\frac{1}{2}\left(-{g^i}^j{{g_t}_t}_{,j}+{g^i}^t{{g_t}_t}_{,t}+2{g^i}^j{{g_j}_t}_{,t}\right)$
replace ADM metric definitions
$\frac{1}{2}\left(-\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,j}+\frac{{\beta }_i}{{\alpha }^2}{(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,t}+2\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{\beta }_j}_{,t}\right)$
raise ${\beta }_j$
$\frac{1}{2}\left(-\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,j}+\frac{{\beta }_i}{{\alpha }^2}{(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,t}+2\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){\left({{\gamma }_j}_k{\beta }^k\right)}_{,t}\right)$
simplify...
$-1{\alpha }_{,t}{\beta }_i\frac{1}{\alpha }+\alpha \cdot {\alpha }_{,j}{{\gamma }^i}^j+\frac{1}{2}{\beta }_i{\beta }^k{{\beta }^l}_{,t}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{\beta }^l{{\beta }^k}_{,t}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,t}\frac{1}{{\alpha }^2}+-1{\alpha }_{,j}{\beta }^i{\beta }^j\frac{1}{\alpha }+-1{\beta }^i{\beta }^j{{\beta }^k}_{,t}{{\gamma }_j}_k\frac{1}{{\alpha }^2}+{\beta }^k{{\gamma }^i}^j{{{\gamma }_j}_k}_{,t}+-1{\beta }^i{\beta }^j{\beta }^k{{{\gamma }_j}_k}_{,t}\frac{1}{{\alpha }^2}+{{\beta }^k}_{,t}{{\gamma }^i}^j{{\gamma }_j}_k+-1\cdot \frac{1}{2}{\beta }^l{{\beta }^k}_{,j}{{\gamma }^i}^j{{\gamma }_k}_l+\frac{1}{2}{\beta }^i{\beta }^j{\beta }^l{{\beta }^k}_{,j}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^k{\beta }^l{{\gamma }^i}^j{{{\gamma }_k}_l}_{,j}+-1\cdot \frac{1}{2}{\beta }^k{{\beta }^l}_{,j}{{\gamma }^i}^j{{\gamma }_k}_l+\frac{1}{2}{\beta }^i{\beta }^j{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,j}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }^i{\beta }^j{\beta }^k{{\beta }^l}_{,j}{{\gamma }_k}_l\frac{1}{{\alpha }^2}$
relabel...
$-1{\alpha }_{,t}{\beta }_i\frac{1}{\alpha }+\alpha \cdot {\alpha }_{,j}{{\gamma }^i}^j+\frac{1}{2}{\beta }_i{\beta }^k{{\beta }^l}_{,t}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{\beta }^l{{\beta }^k}_{,t}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,t}\frac{1}{{\alpha }^2}+-1{\alpha }_{,j}{\beta }^i{\beta }^j\frac{1}{\alpha }+{\beta }^k{{\gamma }^i}^j{{{\gamma }_j}_k}_{,t}+-1{\beta }^i{\beta }^j{\beta }^k{{{\gamma }_j}_k}_{,t}\frac{1}{{\alpha }^2}+-1{\beta }^i{\beta }^j{{\beta }^k}_{,t}{{\gamma }_j}_k\frac{1}{{\alpha }^2}+{{\beta }^k}_{,t}{{\gamma }^i}^j{{\gamma }_j}_k+\frac{1}{2}{\beta }^i{\beta }^j{\beta }^l{{\beta }^k}_{,j}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^k{\beta }^l{{\gamma }^i}^j{{{\gamma }_k}_l}_{,j}+\frac{1}{2}{\beta }^i{\beta }^j{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,j}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^l{{\beta }^k}_{,j}{{\gamma }^i}^j{{\gamma }_k}_l+\frac{1}{2}{\beta }^i{\beta }^j{\beta }^k{{\beta }^l}_{,j}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^k{{\beta }^l}_{,j}{{\gamma }^i}^j{{\gamma }_k}_l$
for ${\beta }^i=0$
$\alpha \cdot {\alpha }_{,j}{{\gamma }^i}^j$

---

${{{\mathrm{\Gamma }}^i}_m}_t$
factor out index raise
${g^i}^a{{{\mathrm{\Gamma }}_a}_m}_t$
substitute definition of connection
$\frac{1}{2}{g^i}^a\left({{g_a}_m}_{,t}+{{g_a}_t}_{,m}-{{g_m}_t}_{,a}\right)$
split the index a into t and j
$\frac{1}{2}\left({g^i}^j{{g_j}_m}_{,t}-{g^i}^j{{g_m}_t}_{,j}+{g^i}^j{{g_j}_t}_{,m}+{g^i}^t{{g_t}_t}_{,m}+{g^i}^t{{g_t}_m}_{,t}-{g^i}^t{{g_m}_t}_{,t}\right)$
replace ADM metric definitions
$\frac{1}{2}\left(\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{{\gamma }_j}_m}_{,t}-\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{{\gamma }_m}_t}_{,j}+\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{\beta }_j}_{,m}+\frac{{\beta }_i}{{\alpha }^2}{(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,m}+\frac{{\beta }_i}{{\alpha }^2}{{{\gamma }_t}_m}_{,t}-\frac{{\beta }_i}{{\alpha }^2}{{{\gamma }_m}_t}_{,t}\right)$
raise ${\beta }_j$
$\frac{1}{2}\left(\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{{\gamma }_j}_m}_{,t}-\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{{\gamma }_m}_t}_{,j}+\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){\left({{\gamma }_j}_k{\beta }^k\right)}_{,m}+\frac{{\beta }_i}{{\alpha }^2}{(-{\alpha }^2+{\beta }^k{\beta }^l{{\gamma }_k}_l)}_{,m}+\frac{{\beta }_i}{{\alpha }^2}{{{\gamma }_t}_m}_{,t}-\frac{{\beta }_i}{{\alpha }^2}{{{\gamma }_m}_t}_{,t}\right)$
simplify...
$-1\cdot \frac{1}{2}{\beta }_i{{{\gamma }_m}_t}_{,t}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{{{\gamma }_t}_m}_{,t}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }^i{\beta }^j{{{\gamma }_m}_t}_{,j}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^i{\beta }^j{{{\gamma }_j}_m}_{,t}\frac{1}{{\alpha }^2}+\frac{1}{2}{{\gamma }^i}^j{{{\gamma }_j}_m}_{,t}+-1\cdot \frac{1}{2}{{\gamma }^i}^j{{{\gamma }_m}_t}_{,j}+-1{\alpha }_{,m}{\beta }_i\frac{1}{\alpha }+\frac{1}{2}{\beta }_i{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,m}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^i{\beta }^j{\beta }^k{{{\gamma }_j}_k}_{,m}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^i{\beta }^j{{\beta }^k}_{,m}{{\gamma }_j}_k\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{\beta }^k{{\beta }^l}_{,m}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{\beta }^l{{\beta }^k}_{,m}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }^k{{\gamma }^i}^j{{{\gamma }_j}_k}_{,m}+\frac{1}{2}{{\beta }^k}_{,m}{{\gamma }^i}^j{{\gamma }_j}_k$
relabel...
$-1\cdot \frac{1}{2}{\beta }_i{{{\gamma }_m}_t}_{,t}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{{{\gamma }_t}_m}_{,t}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^i{\beta }^j{{{\gamma }_j}_m}_{,t}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }^i{\beta }^j{{{\gamma }_m}_t}_{,j}\frac{1}{{\alpha }^2}+\frac{1}{2}{{\gamma }^i}^j{{{\gamma }_j}_m}_{,t}+-1\cdot \frac{1}{2}{{\gamma }^i}^j{{{\gamma }_m}_t}_{,j}+-1{\alpha }_{,m}{\beta }_i\frac{1}{\alpha }+\frac{1}{2}{\beta }_i{\beta }^l{{\beta }^k}_{,m}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }^k{{\gamma }^i}^j{{{\gamma }_j}_k}_{,m}+\frac{1}{2}{\beta }_i{\beta }^k{{\beta }^l}_{,m}{{\gamma }_k}_l\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^i{\beta }^j{\beta }^k{{{\gamma }_j}_k}_{,m}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^i{\beta }^j{{\beta }^k}_{,m}{{\gamma }_j}_k\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{\beta }^k{\beta }^l{{{\gamma }_k}_l}_{,m}\frac{1}{{\alpha }^2}+\frac{1}{2}{{\beta }^k}_{,m}{{\gamma }^i}^j{{\gamma }_j}_k$
for ${\beta }^i=0$
$\frac{1}{2}{{\gamma }^i}^j\left(-{{{\gamma }_m}_t}_{,j}+{{{\gamma }_j}_m}_{,t}\right)$

---

${{{\mathrm{\Gamma }}^t}_i}_m$
factor out index raise
${g^t}^a{{{\mathrm{\Gamma }}_a}_i}_m$
substitute definition of connection
$\frac{1}{2}{g^t}^a\left({{g_a}_i}_{,m}+{{g_a}_m}_{,i}-{{g_i}_m}_{,a}\right)$
split the index a into t and j
$\frac{1}{2}\left({g^t}^j{{g_j}_i}_{,m}-{g^t}^j{{g_i}_m}_{,j}+{g^t}^j{{g_j}_m}_{,i}+{g^t}^t{{g_t}_m}_{,i}+{g^t}^t{{g_t}_i}_{,m}-{g^t}^t{{g_i}_m}_{,t}\right)$
replace ADM metric definitions
$\frac{1}{2}\left(\frac{{\beta }_j}{{\alpha }^2}{{{\gamma }_j}_i}_{,m}-\frac{{\beta }_j}{{\alpha }^2}{{{\gamma }_i}_m}_{,j}+\frac{{\beta }_j}{{\alpha }^2}{{{\gamma }_j}_m}_{,i}+{{{\gamma }_t}_m}_{,i}\frac{-1}{{\alpha }^2}+{{{\gamma }_t}_i}_{,m}\frac{-1}{{\alpha }^2}-{{{\gamma }_i}_m}_{,t}\frac{-1}{{\alpha }^2}\right)$
raise ${\beta }_j$
$\frac{1}{2}\left(\frac{{\beta }_j}{{\alpha }^2}{{{\gamma }_j}_i}_{,m}-\frac{{\beta }_j}{{\alpha }^2}{{{\gamma }_i}_m}_{,j}+\frac{{\beta }_j}{{\alpha }^2}{{{\gamma }_j}_m}_{,i}+{{{\gamma }_t}_m}_{,i}\frac{-1}{{\alpha }^2}+{{{\gamma }_t}_i}_{,m}\frac{-1}{{\alpha }^2}-{{{\gamma }_i}_m}_{,t}\frac{-1}{{\alpha }^2}\right)$
simplify...
$\frac{1}{2}{{{\gamma }_i}_m}_{,t}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{{{\gamma }_t}_i}_{,m}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{{{\gamma }_t}_m}_{,i}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_j{{{\gamma }_j}_m}_{,i}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }_j{{{\gamma }_i}_m}_{,j}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_j{{{\gamma }_j}_i}_{,m}\frac{1}{{\alpha }^2}$
relabel...
$\frac{1}{2}{{{\gamma }_i}_m}_{,t}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{{{\gamma }_t}_m}_{,i}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{{{\gamma }_t}_i}_{,m}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_j{{{\gamma }_j}_m}_{,i}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }_j{{{\gamma }_i}_m}_{,j}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_j{{{\gamma }_j}_i}_{,m}\frac{1}{{\alpha }^2}$
for ${\beta }^i=0$
$\frac{-{{{\gamma }_t}_i}_{,m}-{{{\gamma }_t}_m}_{,i}+{{{\gamma }_i}_m}_{,t}}{2{\alpha }^2}$

---

${{{\mathrm{\Gamma }}^i}_m}_n$
factor out index raise
${g^i}^a{{{\mathrm{\Gamma }}_a}_m}_n$
substitute definition of connection
$\frac{1}{2}{g^i}^a\left({{g_a}_m}_{,n}+{{g_a}_n}_{,m}-{{g_m}_n}_{,a}\right)$
split the index a into t and j
$\frac{1}{2}\left({g^i}^j{{g_j}_m}_{,n}-{g^i}^j{{g_m}_n}_{,j}+{g^i}^j{{g_j}_n}_{,m}+{g^i}^t{{g_t}_n}_{,m}+{g^i}^t{{g_t}_m}_{,n}-{g^i}^t{{g_m}_n}_{,t}\right)$
replace ADM metric definitions
$\frac{1}{2}\left(\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{{\gamma }_j}_m}_{,n}-\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{{\gamma }_m}_n}_{,j}+\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{{\gamma }_j}_n}_{,m}+\frac{{\beta }_i}{{\alpha }^2}{{{\gamma }_t}_n}_{,m}+\frac{{\beta }_i}{{\alpha }^2}{{{\gamma }_t}_m}_{,n}-\frac{{\beta }_i}{{\alpha }^2}{{{\gamma }_m}_n}_{,t}\right)$
raise ${\beta }_j$
$\frac{1}{2}\left(\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{{\gamma }_j}_m}_{,n}-\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{{\gamma }_m}_n}_{,j}+\left({{\gamma }^i}^j-\frac{{\beta }^i{\beta }^j}{{\alpha }^2}\right){{{\gamma }_j}_n}_{,m}+\frac{{\beta }_i}{{\alpha }^2}{{{\gamma }_t}_n}_{,m}+\frac{{\beta }_i}{{\alpha }^2}{{{\gamma }_t}_m}_{,n}-\frac{{\beta }_i}{{\alpha }^2}{{{\gamma }_m}_n}_{,t}\right)$
simplify...
$-1\cdot \frac{1}{2}{\beta }_i{{{\gamma }_m}_n}_{,t}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{{{\gamma }_t}_n}_{,m}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{{{\gamma }_t}_m}_{,n}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }^i{\beta }^j{{{\gamma }_m}_n}_{,j}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^i{\beta }^j{{{\gamma }_j}_n}_{,m}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^i{\beta }^j{{{\gamma }_j}_m}_{,n}\frac{1}{{\alpha }^2}+\frac{1}{2}{{\gamma }^i}^j{{{\gamma }_j}_m}_{,n}+\frac{1}{2}{{\gamma }^i}^j{{{\gamma }_j}_n}_{,m}+-1\cdot \frac{1}{2}{{\gamma }^i}^j{{{\gamma }_m}_n}_{,j}$
relabel...
$-1\cdot \frac{1}{2}{\beta }_i{{{\gamma }_m}_n}_{,t}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{{{\gamma }_t}_m}_{,n}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }_i{{{\gamma }_t}_n}_{,m}\frac{1}{{\alpha }^2}+\frac{1}{2}{\beta }^i{\beta }^j{{{\gamma }_m}_n}_{,j}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^i{\beta }^j{{{\gamma }_j}_n}_{,m}\frac{1}{{\alpha }^2}+-1\cdot \frac{1}{2}{\beta }^i{\beta }^j{{{\gamma }_j}_m}_{,n}\frac{1}{{\alpha }^2}+\frac{1}{2}{{\gamma }^i}^j{{{\gamma }_j}_m}_{,n}+\frac{1}{2}{{\gamma }^i}^j{{{\gamma }_j}_n}_{,m}+-1\cdot \frac{1}{2}{{\gamma }^i}^j{{{\gamma }_m}_n}_{,j}$
for ${\beta }^i=0$
$\frac{1}{2}{{\gamma }^i}^j\left(-{{{\gamma }_m}_n}_{,j}+{{{\gamma }_j}_n}_{,m}+{{{\gamma }_j}_m}_{,n}\right)$

---

${{{R^t}_t}_t}_i$
substitute definition of Riemann curvature
${{{{\mathrm{\Gamma }}^t}_t}_i}_{,t}-{{{{\mathrm{\Gamma }}^t}_t}_t}_{,i}+{{{\mathrm{\Gamma }}^t}_e}_t{{{\mathrm{\Gamma }}^e}_t}_i-{{{\mathrm{\Gamma }}^t}_e}_i{{{\mathrm{\Gamma }}^e}_t}_t$
split the index e into t and m
${{{{\mathrm{\Gamma }}^t}_t}_i}_{,t}-{{{{\mathrm{\Gamma }}^t}_t}_t}_{,i}+\left({{{\mathrm{\Gamma }}^t}_t}_t+{{{\mathrm{\Gamma }}^t}_m}_t\right)\left({{{\mathrm{\Gamma }}^t}_t}_i+{{{\mathrm{\Gamma }}^m}_t}_i\right)-\left({{{\mathrm{\Gamma }}^t}_t}_i+{{{\mathrm{\Gamma }}^t}_n}_i\right)\left({{{\mathrm{\Gamma }}^t}_t}_t+{{{\mathrm{\Gamma }}^n}_t}_t\right)$
${{{{\mathrm{\Gamma }}^t}_t}_i}_{,t}-{{{{\mathrm{\Gamma }}^t}_t}_t}_{,i}+{{{\mathrm{\Gamma }}^t}_m}_t{{{\mathrm{\Gamma }}^t}_t}_i+{{{\mathrm{\Gamma }}^m}_t}_i{{{\mathrm{\Gamma }}^t}_t}_t+{{{\mathrm{\Gamma }}^m}_t}_i{{{\mathrm{\Gamma }}^t}_m}_t-{{{\mathrm{\Gamma }}^t}_n}_i{{{\mathrm{\Gamma }}^t}_t}_t-{{{\mathrm{\Gamma }}^n}_t}_t{{{\mathrm{\Gamma }}^t}_t}_i-{{{\mathrm{\Gamma }}^n}_t}_t{{{\mathrm{\Gamma }}^t}_n}_i$

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${{{R^t}_t}_i}_j$
substitute definition of Riemann curvature
${{{{\mathrm{\Gamma }}^t}_t}_j}_{,i}-{{{{\mathrm{\Gamma }}^t}_t}_i}_{,j}+{{{\mathrm{\Gamma }}^t}_e}_i{{{\mathrm{\Gamma }}^e}_t}_j-{{{\mathrm{\Gamma }}^t}_e}_j{{{\mathrm{\Gamma }}^e}_t}_i$
split the index e into t and m
${{{{\mathrm{\Gamma }}^t}_t}_j}_{,i}-{{{{\mathrm{\Gamma }}^t}_t}_i}_{,j}+\left({{{\mathrm{\Gamma }}^t}_t}_i+{{{\mathrm{\Gamma }}^t}_m}_i\right)\left({{{\mathrm{\Gamma }}^t}_t}_j+{{{\mathrm{\Gamma }}^m}_t}_j\right)-\left({{{\mathrm{\Gamma }}^t}_t}_j+{{{\mathrm{\Gamma }}^t}_n}_j\right)\left({{{\mathrm{\Gamma }}^t}_t}_i+{{{\mathrm{\Gamma }}^n}_t}_i\right)$
${{{{\mathrm{\Gamma }}^t}_t}_j}_{,i}-{{{{\mathrm{\Gamma }}^t}_t}_i}_{,j}+{{{\mathrm{\Gamma }}^t}_m}_i{{{\mathrm{\Gamma }}^t}_t}_j+{{{\mathrm{\Gamma }}^m}_t}_j{{{\mathrm{\Gamma }}^t}_t}_i+{{{\mathrm{\Gamma }}^m}_t}_j{{{\mathrm{\Gamma }}^t}_m}_i-{{{\mathrm{\Gamma }}^t}_n}_j{{{\mathrm{\Gamma }}^t}_t}_i-{{{\mathrm{\Gamma }}^n}_t}_i{{{\mathrm{\Gamma }}^t}_t}_j-{{{\mathrm{\Gamma }}^n}_t}_i{{{\mathrm{\Gamma }}^t}_n}_j$

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${{{R^t}_i}_t}_j$
substitute definition of Riemann curvature
${{{{\mathrm{\Gamma }}^t}_i}_j}_{,t}-{{{{\mathrm{\Gamma }}^t}_i}_t}_{,j}+{{{\mathrm{\Gamma }}^t}_e}_t{{{\mathrm{\Gamma }}^e}_i}_j-{{{\mathrm{\Gamma }}^t}_e}_j{{{\mathrm{\Gamma }}^e}_i}_t$
split the index e into t and m
${{{{\mathrm{\Gamma }}^t}_i}_j}_{,t}-{{{{\mathrm{\Gamma }}^t}_i}_t}_{,j}+\left({{{\mathrm{\Gamma }}^t}_t}_t+{{{\mathrm{\Gamma }}^t}_m}_t\right)\left({{{\mathrm{\Gamma }}^t}_i}_j+{{{\mathrm{\Gamma }}^m}_i}_j\right)-\left({{{\mathrm{\Gamma }}^t}_t}_j+{{{\mathrm{\Gamma }}^t}_n}_j\right)\left({{{\mathrm{\Gamma }}^t}_i}_t+{{{\mathrm{\Gamma }}^n}_i}_t\right)$
${{{{\mathrm{\Gamma }}^t}_i}_j}_{,t}-{{{{\mathrm{\Gamma }}^t}_i}_t}_{,j}+{{{\mathrm{\Gamma }}^t}_i}_j{{{\mathrm{\Gamma }}^t}_m}_t-{{{\mathrm{\Gamma }}^n}_i}_t{{{\mathrm{\Gamma }}^t}_t}_j+{{{\mathrm{\Gamma }}^m}_i}_j{{{\mathrm{\Gamma }}^t}_t}_t+{{{\mathrm{\Gamma }}^m}_i}_j{{{\mathrm{\Gamma }}^t}_m}_t-{{{\mathrm{\Gamma }}^t}_i}_t{{{\mathrm{\Gamma }}^t}_n}_j+{{{\mathrm{\Gamma }}^t}_i}_j{{{\mathrm{\Gamma }}^t}_t}_t-{{{\mathrm{\Gamma }}^t}_i}_t{{{\mathrm{\Gamma }}^t}_t}_j-{{{\mathrm{\Gamma }}^n}_i}_t{{{\mathrm{\Gamma }}^t}_n}_j$

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${{{R^i}_t}_t}_j$
substitute definition of Riemann curvature
${{{{\mathrm{\Gamma }}^i}_t}_j}_{,t}-{{{{\mathrm{\Gamma }}^i}_t}_t}_{,j}+{{{\mathrm{\Gamma }}^i}_e}_t{{{\mathrm{\Gamma }}^e}_t}_j-{{{\mathrm{\Gamma }}^i}_e}_j{{{\mathrm{\Gamma }}^e}_t}_t$
split the index e into t and m
${{{{\mathrm{\Gamma }}^i}_t}_j}_{,t}-{{{{\mathrm{\Gamma }}^i}_t}_t}_{,j}+\left({{{\mathrm{\Gamma }}^i}_t}_t+{{{\mathrm{\Gamma }}^i}_m}_t\right)\left({{{\mathrm{\Gamma }}^t}_t}_j+{{{\mathrm{\Gamma }}^m}_t}_j\right)-\left({{{\mathrm{\Gamma }}^i}_t}_j+{{{\mathrm{\Gamma }}^i}_n}_j\right)\left({{{\mathrm{\Gamma }}^t}_t}_t+{{{\mathrm{\Gamma }}^n}_t}_t\right)$
${{{{\mathrm{\Gamma }}^i}_t}_j}_{,t}-{{{{\mathrm{\Gamma }}^i}_t}_t}_{,j}+{{{\mathrm{\Gamma }}^i}_m}_t{{{\mathrm{\Gamma }}^t}_t}_j-{{{\mathrm{\Gamma }}^i}_t}_j{{{\mathrm{\Gamma }}^n}_t}_t+{{{\mathrm{\Gamma }}^i}_t}_t{{{\mathrm{\Gamma }}^m}_t}_j+{{{\mathrm{\Gamma }}^i}_m}_t{{{\mathrm{\Gamma }}^m}_t}_j-{{{\mathrm{\Gamma }}^i}_n}_j{{{\mathrm{\Gamma }}^t}_t}_t+{{{\mathrm{\Gamma }}^i}_t}_t{{{\mathrm{\Gamma }}^t}_t}_j-{{{\mathrm{\Gamma }}^i}_t}_j{{{\mathrm{\Gamma }}^t}_t}_t-{{{\mathrm{\Gamma }}^i}_n}_j{{{\mathrm{\Gamma }}^n}_t}_t$

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${{{R^t}_i}_j}_k$
substitute definition of Riemann curvature
${{{{\mathrm{\Gamma }}^t}_i}_k}_{,j}-{{{{\mathrm{\Gamma }}^t}_i}_j}_{,k}+{{{\mathrm{\Gamma }}^t}_e}_j{{{\mathrm{\Gamma }}^e}_i}_k-{{{\mathrm{\Gamma }}^t}_e}_k{{{\mathrm{\Gamma }}^e}_i}_j$
split the index e into t and m
${{{{\mathrm{\Gamma }}^t}_i}_k}_{,j}-{{{{\mathrm{\Gamma }}^t}_i}_j}_{,k}+\left({{{\mathrm{\Gamma }}^t}_t}_j+{{{\mathrm{\Gamma }}^t}_m}_j\right)\left({{{\mathrm{\Gamma }}^t}_i}_k+{{{\mathrm{\Gamma }}^m}_i}_k\right)-\left({{{\mathrm{\Gamma }}^t}_t}_k+{{{\mathrm{\Gamma }}^t}_n}_k\right)\left({{{\mathrm{\Gamma }}^t}_i}_j+{{{\mathrm{\Gamma }}^n}_i}_j\right)$
${{{{\mathrm{\Gamma }}^t}_i}_k}_{,j}-{{{{\mathrm{\Gamma }}^t}_i}_j}_{,k}+{{{\mathrm{\Gamma }}^t}_i}_k{{{\mathrm{\Gamma }}^t}_m}_j-{{{\mathrm{\Gamma }}^n}_i}_j{{{\mathrm{\Gamma }}^t}_t}_k+{{{\mathrm{\Gamma }}^m}_i}_k{{{\mathrm{\Gamma }}^t}_t}_j+{{{\mathrm{\Gamma }}^m}_i}_k{{{\mathrm{\Gamma }}^t}_m}_j-{{{\mathrm{\Gamma }}^t}_i}_j{{{\mathrm{\Gamma }}^t}_n}_k+{{{\mathrm{\Gamma }}^t}_i}_k{{{\mathrm{\Gamma }}^t}_t}_j-{{{\mathrm{\Gamma }}^t}_i}_j{{{\mathrm{\Gamma }}^t}_t}_k-{{{\mathrm{\Gamma }}^n}_i}_j{{{\mathrm{\Gamma }}^t}_n}_k$

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${{{R^i}_t}_j}_k$
substitute definition of Riemann curvature
${{{{\mathrm{\Gamma }}^i}_t}_k}_{,j}-{{{{\mathrm{\Gamma }}^i}_t}_j}_{,k}+{{{\mathrm{\Gamma }}^i}_e}_j{{{\mathrm{\Gamma }}^e}_t}_k-{{{\mathrm{\Gamma }}^i}_e}_k{{{\mathrm{\Gamma }}^e}_t}_j$
split the index e into t and m
${{{{\mathrm{\Gamma }}^i}_t}_k}_{,j}-{{{{\mathrm{\Gamma }}^i}_t}_j}_{,k}+\left({{{\mathrm{\Gamma }}^i}_t}_j+{{{\mathrm{\Gamma }}^i}_m}_j\right)\left({{{\mathrm{\Gamma }}^t}_t}_k+{{{\mathrm{\Gamma }}^m}_t}_k\right)-\left({{{\mathrm{\Gamma }}^i}_t}_k+{{{\mathrm{\Gamma }}^i}_n}_k\right)\left({{{\mathrm{\Gamma }}^t}_t}_j+{{{\mathrm{\Gamma }}^n}_t}_j\right)$
${{{{\mathrm{\Gamma }}^i}_t}_k}_{,j}-{{{{\mathrm{\Gamma }}^i}_t}_j}_{,k}+{{{\mathrm{\Gamma }}^i}_m}_j{{{\mathrm{\Gamma }}^t}_t}_k-{{{\mathrm{\Gamma }}^i}_t}_k{{{\mathrm{\Gamma }}^n}_t}_j+{{{\mathrm{\Gamma }}^i}_t}_j{{{\mathrm{\Gamma }}^m}_t}_k+{{{\mathrm{\Gamma }}^i}_m}_j{{{\mathrm{\Gamma }}^m}_t}_k-{{{\mathrm{\Gamma }}^i}_n}_k{{{\mathrm{\Gamma }}^t}_t}_j+{{{\mathrm{\Gamma }}^i}_t}_j{{{\mathrm{\Gamma }}^t}_t}_k-{{{\mathrm{\Gamma }}^i}_t}_k{{{\mathrm{\Gamma }}^t}_t}_j-{{{\mathrm{\Gamma }}^i}_n}_k{{{\mathrm{\Gamma }}^n}_t}_j$

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${{{R^i}_j}_t}_k$
substitute definition of Riemann curvature
${{{{\mathrm{\Gamma }}^i}_j}_k}_{,t}-{{{{\mathrm{\Gamma }}^i}_j}_t}_{,k}+{{{\mathrm{\Gamma }}^i}_e}_t{{{\mathrm{\Gamma }}^e}_j}_k-{{{\mathrm{\Gamma }}^i}_e}_k{{{\mathrm{\Gamma }}^e}_j}_t$
split the index e into t and m
${{{{\mathrm{\Gamma }}^i}_j}_k}_{,t}-{{{{\mathrm{\Gamma }}^i}_j}_t}_{,k}+\left({{{\mathrm{\Gamma }}^i}_t}_t+{{{\mathrm{\Gamma }}^i}_m}_t\right)\left({{{\mathrm{\Gamma }}^t}_j}_k+{{{\mathrm{\Gamma }}^m}_j}_k\right)-\left({{{\mathrm{\Gamma }}^i}_t}_k+{{{\mathrm{\Gamma }}^i}_n}_k\right)\left({{{\mathrm{\Gamma }}^t}_j}_t+{{{\mathrm{\Gamma }}^n}_j}_t\right)$
${{{{\mathrm{\Gamma }}^i}_j}_k}_{,t}-{{{{\mathrm{\Gamma }}^i}_j}_t}_{,k}+{{{\mathrm{\Gamma }}^i}_m}_t{{{\mathrm{\Gamma }}^t}_j}_k-{{{\mathrm{\Gamma }}^i}_t}_k{{{\mathrm{\Gamma }}^n}_j}_t+{{{\mathrm{\Gamma }}^i}_t}_t{{{\mathrm{\Gamma }}^m}_j}_k+{{{\mathrm{\Gamma }}^i}_m}_t{{{\mathrm{\Gamma }}^m}_j}_k-{{{\mathrm{\Gamma }}^i}_n}_k{{{\mathrm{\Gamma }}^t}_j}_t+{{{\mathrm{\Gamma }}^i}_t}_t{{{\mathrm{\Gamma }}^t}_j}_k-{{{\mathrm{\Gamma }}^i}_t}_k{{{\mathrm{\Gamma }}^t}_j}_t-{{{\mathrm{\Gamma }}^i}_n}_k{{{\mathrm{\Gamma }}^n}_j}_t$

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${{{R^i}_j}_k}_l$
substitute definition of Riemann curvature
${{{{\mathrm{\Gamma }}^i}_j}_l}_{,k}-{{{{\mathrm{\Gamma }}^i}_j}_k}_{,l}+{{{\mathrm{\Gamma }}^i}_e}_k{{{\mathrm{\Gamma }}^e}_j}_l-{{{\mathrm{\Gamma }}^i}_e}_l{{{\mathrm{\Gamma }}^e}_j}_k$
split the index e into t and m
${{{{\mathrm{\Gamma }}^i}_j}_l}_{,k}-{{{{\mathrm{\Gamma }}^i}_j}_k}_{,l}+\left({{{\mathrm{\Gamma }}^i}_t}_k+{{{\mathrm{\Gamma }}^i}_m}_k\right)\left({{{\mathrm{\Gamma }}^t}_j}_l+{{{\mathrm{\Gamma }}^m}_j}_l\right)-\left({{{\mathrm{\Gamma }}^i}_t}_l+{{{\mathrm{\Gamma }}^i}_n}_l\right)\left({{{\mathrm{\Gamma }}^t}_j}_k+{{{\mathrm{\Gamma }}^n}_j}_k\right)$
${{{{\mathrm{\Gamma }}^i}_j}_l}_{,k}-{{{{\mathrm{\Gamma }}^i}_j}_k}_{,l}+{{{\mathrm{\Gamma }}^i}_m}_k{{{\mathrm{\Gamma }}^t}_j}_l-{{{\mathrm{\Gamma }}^i}_t}_l{{{\mathrm{\Gamma }}^n}_j}_k+{{{\mathrm{\Gamma }}^i}_t}_k{{{\mathrm{\Gamma }}^m}_j}_l+{{{\mathrm{\Gamma }}^i}_m}_k{{{\mathrm{\Gamma }}^m}_j}_l-{{{\mathrm{\Gamma }}^i}_n}_l{{{\mathrm{\Gamma }}^t}_j}_k+{{{\mathrm{\Gamma }}^i}_t}_k{{{\mathrm{\Gamma }}^t}_j}_l-{{{\mathrm{\Gamma }}^i}_t}_l{{{\mathrm{\Gamma }}^t}_j}_k-{{{\mathrm{\Gamma }}^i}_n}_l{{{\mathrm{\Gamma }}^n}_j}_k$