Now introduce $\Gamma^k$ instead
$\Gamma^k = {\Gamma^k}_{ij} \gamma^{ij}$
$= \gamma^{kl} \Gamma_{lij} \gamma^{ij}$
$= \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij}$
$= 2 {d_i}^{ik} - {d^{ki}}_i$
$= {d_i}^{ik} - V^k$
so
$V^k = {d_i}^{ik} - \Gamma^k$

${\Gamma^k}_{,\mu}$
$= (\gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij})_{,\mu}$
$= \gamma^{kl}_{,\mu} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij} + \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij})_{,\mu} \gamma^{ij} + \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij}_{,\mu} $
$= \gamma^{kl} \gamma^{ij} (2 d_{ilj,\mu} - d_{lij,\mu}) - \gamma^{km} \gamma_{mn,\mu} \gamma^{nl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij} - \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{im} \gamma_{mn,\mu} \gamma^{nj}$
$= \gamma^{kl} \gamma^{ij} (2 d_{ijl,\mu} - d_{lij,\mu}) - \gamma^{km} \gamma_{mn,\mu} \Gamma^n - (2 d^{mnk} - d^{kmn}) \gamma_{mn,\mu}$

${\Gamma^k}_{,r} = \gamma^{kl} \gamma^{ij} (2 d_{ilj,r} - d_{lij,r}) - \gamma^{km} \gamma_{mn,r} \gamma^{nl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij} - \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{im} \gamma_{mn,r} \gamma^{nj}$
$= (2 \gamma^{kp} \gamma^{mq} - \gamma^{km} \gamma^{pq}) d_{mpq,r} - 2 {d_r}^{kj} \Gamma_j - 2 {d_r}^{ij} (2 {d_{ij}}^k - {d^k}_{ij}) $

${\Gamma^k}_{,t} = \gamma^{kl} \gamma^{ij} (2 d_{ilj,t} - d_{lij,t}) - \gamma^{km} \gamma_{mn,t} \gamma^{nl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{ij} - \gamma^{kl} (d_{jli} + d_{ilj} - d_{lij}) \gamma^{im} \gamma_{mn,t} \gamma^{nj} $
$ = \gamma^{kl} \gamma^{ij} ( 2 \delta^m_i \delta^p_l \delta^q_j - \delta^m_l \delta^p_i \delta^q_j ) d_{mpq,t} - (d_{jli} + d_{ilj} - d_{lij}) ( \gamma^{km} \gamma^{nl} \gamma^{ij} + \gamma^{kl} \gamma^{im} \gamma^{nj} ) \gamma_{mn,t} $
$ = ( 2 \gamma^{kp} \gamma^{mq} - \gamma^{km} \gamma^{pq} ) ( d_{mpq,l} \beta^l - \alpha K_{pq,m} + d_{lpq} {\beta^l}_{,m} + 2 d_{ml(p} {\beta^l}_{,q)} + \gamma_{l(p} {\beta^l}_{,q)m} - \alpha a_m K_{pq} ) - (d_{jli} + d_{ilj} - d_{lij}) ( \gamma^{km} \gamma^{nl} \gamma^{ij} + \gamma^{kl} \gamma^{im} \gamma^{nj} ) ( \gamma_{mn,p} \beta^p + 2 \gamma_{p(m} {\beta^p}_{,n)} - 2 \alpha K_{mn} )$
$ = 2 \gamma^{kp} \gamma^{mq} d_{mpq,l} \beta^l - \gamma^{km} \gamma^{pq} d_{mpq,l} \beta^l - 2 \gamma^{kp} \gamma^{mq} \alpha K_{pq,m} + \gamma^{km} \gamma^{pq} \alpha K_{pq,m} + 2 \gamma^{kp} \gamma^{mq} d_{lpq} {\beta^l}_{,m} - \gamma^{km} \gamma^{pq} d_{lpq} {\beta^l}_{,m} + 2 \gamma^{kp} \gamma^{mq} 2 d_{ml(p} {\beta^l}_{,q)} - \gamma^{km} \gamma^{pq} 2 d_{ml(p} {\beta^l}_{,q)} + 2 \gamma^{kp} \gamma^{mq} \gamma_{l(p} {\beta^l}_{,q)m} - \gamma^{km} \gamma^{pq} \gamma_{l(p} {\beta^l}_{,q)m} - 2 \gamma^{kp} \gamma^{mq} \alpha a_m K_{pq} + \gamma^{km} \gamma^{pq} \alpha a_m K_{pq} - d_{jli} \gamma^{km} \gamma^{nl} \gamma^{ij} \gamma_{mn,p} \beta^p - d_{jli} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \gamma_{p(m} {\beta^p}_{,n)} + d_{jli} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \alpha K_{mn} - d_{ilj} \gamma^{km} \gamma^{nl} \gamma^{ij} \gamma_{mn,p} \beta^p - d_{ilj} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \gamma_{p(m} {\beta^p}_{,n)} + d_{ilj} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \alpha K_{mn} + d_{lij} \gamma^{km} \gamma^{nl} \gamma^{ij} \gamma_{mn,p} \beta^p + d_{lij} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \gamma_{p(m} {\beta^p}_{,n)} - d_{lij} \gamma^{km} \gamma^{nl} \gamma^{ij} 2 \alpha K_{mn} - d_{jli} \gamma^{kl} \gamma^{im} \gamma^{nj} \gamma_{mn,p} \beta^p - d_{jli} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \gamma_{p(m} {\beta^p}_{,n)} + d_{jli} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \alpha K_{mn} - d_{ilj} \gamma^{kl} \gamma^{im} \gamma^{nj} \gamma_{mn,p} \beta^p - d_{ilj} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \gamma_{p(m} {\beta^p}_{,n)} + d_{ilj} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \alpha K_{mn} + d_{lij} \gamma^{kl} \gamma^{im} \gamma^{nj} \gamma_{mn,p} \beta^p + d_{lij} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \gamma_{p(m} {\beta^p}_{,n)} - d_{lij} \gamma^{kl} \gamma^{im} \gamma^{nj} 2 \alpha K_{mn} $
$ = (2 \gamma^{kp} \gamma^{mq} - \gamma^{km} \gamma^{pq}) \beta^r d_{mpq,r} - \alpha (2 \gamma^{kp} \gamma^{qr} - \gamma^{kr} \gamma^{pq}) K_{pq,r} - 2 ({d_l}^{kr} + {d^{rk}}_l) {\beta^l}_{,r} - (2 {d_i}^{ir} - {d^{ri}}_i) \delta^k_l {\beta^l}_{,r} + {\beta^k}_{,ij} \gamma^{ij} - 2 {d_l}^{jk} (2 {d^i}_{ij} - {d_{ji}}^i) \beta^l - 2 d_{lij} (2 d^{ijk} - d^{kij}) \beta^l - 2 \alpha a_i K^{ki} + \alpha a^k K + 4 \alpha {d_i}^{il} {K^k}_l - 2 \alpha {d^{li}}_i {K^k}_l + 4 \alpha d^{ijk} K_{ij} - 2 \alpha d^{kij} K_{ij} $

...and for the lower form
$\Gamma_k = \Gamma_{kij} \gamma^{ij}$
$= (d_{jki} + d_{ikj} - d_{kij}) \gamma^{ij}$
$= 2 {d^i}_{ik} - {d_{ki}}^i$
$= {d^i}_{ik} - V_k$

${\Gamma_k}_{,\mu}$
$= {\gamma_{ki} \Gamma^i}_{,\mu}$
$= \gamma_{ki,\mu} \Gamma^i + \gamma_{ki} \Gamma^{i,\mu}$

$\Gamma_{k,r} = \gamma_{ki,r} \Gamma^i + \gamma_{ki} {\Gamma^i}_{,r}$
$= 2 d_{rki} \Gamma^i + \gamma_{ki} ( (2 \gamma^{ip} \gamma^{mq} - \gamma^{im} \gamma^{pq}) d_{mpq,r} - 2 {d_r}^{ij} \Gamma_j - 2 {d_r}^{kj} (2 {d_{kj}}^i - {d^i}_{kj}) )$
$= 2 d_{rki} \Gamma^i + (2 \delta^p_k \gamma^{mq} - \delta^m_k \gamma^{pq}) d_{mpq,r} - 2 d_{rkj} \Gamma^j - 2 {d_r}^{ij} (2 d_{ijk} - d_{kij}) $

$\Gamma_{k,t} = (\gamma_{ki} \Gamma^i)_{,t}$
$= \gamma_{ki,t} \Gamma^i + \gamma_{ki} {\Gamma^i}_{,t}$
$= ( \gamma_{ik,j} \beta^j + \gamma_{jk} {\beta^j}_{,i} + \gamma_{ij} {\beta^j}_{,k} - 2 \alpha K_{ik} ) \Gamma^i + \gamma_{ki} ( (2 \gamma^{ip} \gamma^{mq} - \gamma^{im} \gamma^{pq}) \beta^r d_{mpq,r} - \alpha (2 \gamma^{ip} \gamma^{qr} - \gamma^{ir} \gamma^{pq}) K_{pq,r} - 2 ({d_l}^{ir} + {d^{ri}}_l) {\beta^l}_{,r} - (2 {d_n}^{nr} - {d^{rn}}_n) \delta^i_l {\beta^l}_{,r} + {\beta^i}_{,nj} \gamma^{nj} - 2 {d_l}^{ji} (2 {d^n}_{nj} - {d_{jn}}^n) \beta^l - 2 d_{lnj} (2 d^{nji} - d^{inj}) \beta^l - 2 \alpha a_n K^{in} + \alpha a^i K + 4 \alpha {d_n}^{nl} {K^i}_l - 2 \alpha {d^{ln}}_n {K^i}_l + 4 \alpha d^{nji} K_{nj} - 2 \alpha d^{inj} K_{nj} )$
$= + \gamma_{jk} {\beta^j}_{,i} \Gamma^i + \gamma_{ij} {\beta^j}_{,k} \Gamma^i - 2 \alpha K_{ik} \Gamma^i + ( 2 d_{rki} \Gamma^i + (2 \delta^p_k \gamma^{mq} - \delta^m_k \gamma^{pq}) d_{mpq,r} - 2 d_{rkj} \Gamma^j - 2 {d_r}^{ij} (2 d_{ijk} - d_{kij}) ) \beta^r - \alpha (2 \delta^p_k \gamma^{qr} - \delta^r_k \gamma^{pq}) K_{pq,r} - 2 ({d_{lk}}^r + {d^r}_{kl}) {\beta^l}_{,r} - (2 {d_n}^{nr} - {d^{rn}}_n) \gamma_{kl} {\beta^l}_{,r} + \gamma_{kl} {\beta^l}_{,ij} \gamma^{ij} - 2 \alpha a^i K_{ki} + \alpha a_k K + 4 \alpha {d_i}^{ij} K_{kj} - 2 \alpha {d^{ji}}_i K_{kj} + 4 \alpha {d^{ij}}_k K_{ij} - 2 \alpha {d_k}^{ij} K_{ij} $
$= - \alpha (2 \delta^p_k \gamma^{qr} - \delta^r_k \gamma^{pq}) K_{pq,r} - 2 ({d_{lk}}^r + {d^r}_{kl}) {\beta^l}_{,r} - (2 {d_n}^{nr} - {d^{rn}}_n) \gamma_{kl} {\beta^l}_{,r} + \delta^m_k \beta^r \Gamma_{m,r} + \gamma_{kl} \Gamma^r {\beta^l}_{,r} + \delta^r_k \Gamma_l {\beta^l}_{,r} + \gamma_{kl} {\beta^l}_{,ij} \gamma^{ij} - 2 \alpha K_{ik} \Gamma^i - 2 \alpha a^i K_{ki} + \alpha a_k K + 4 \alpha {d_i}^{ij} K_{kj} - 2 \alpha {d^{ji}}_i K_{kj} + 4 \alpha {d^{ij}}_k K_{ij} - 2 \alpha {d_k}^{ij} K_{ij} $

neglecting shift...
$ = - 2 \gamma^{kp} \gamma^{qr} \alpha K_{pq,r} + \gamma^{kr} \gamma^{pq} \alpha K_{pq,r} - 2 \alpha a_i K^{ki} + \alpha a^k K + 2 \alpha (2 {d_i}^{il} - {d^{li}}_i) {K^k}_l + 2 \alpha (2 d^{ijk} - d^{kij}) K_{ij} $
neglecting source...
$ = \alpha (\gamma^{kr} \gamma^{pq} - 2 \gamma^{kp} \gamma^{qr}) K_{pq,r}$

Now for application to $K_{ij,t}$...
$K_{ij,t} = - \alpha \delta^m_{(i} \delta^r_{j)} a_{m,r} + \alpha ( \gamma^{pq} \delta^m_{(i} \delta^r_{j)} - \gamma^{mr} \delta^p_{(i} \delta^q_{j)} ) d_{mpq,r} - 2 \alpha \delta^m_{(i} \delta^r_{j)} V_{m,r} + \beta^k K_{(ij),k} + 2 K_{k(i} {\beta^k}_{,j)} + \alpha ( - a_{(i} a_{j)} + (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k + V_k - {d^l}_{lk}) + 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kli} + 4 {d_{(i}}^{kl} d_{kl|j)} - 3 {d_{(i}}^{kl} d_{j)kl} + K K_{(ij)} - 2 K_{(i|k} {K^k}_{j)} ) + 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij}) $
$= - \alpha \delta^m_{(i} \delta^r_{j)} a_{m,r} + \alpha ( \gamma^{pq} \delta^m_{(i} \delta^r_{j)} - \gamma^{mr} \delta^p_{(i} \delta^q_{j)} ) d_{mpq,r} - 2 \alpha \delta^m_{(i} \delta^r_{j)} ({d^k}_{km} - \Gamma_m)_{,r} + \beta^k K_{(ij),k} + 2 K_{k(i} {\beta^k}_{,j)} + \alpha ( - a_{(i} a_{j)} + (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k + {d^l}_{lk} - \Gamma_k - {d^l}_{lk}) + 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kli} + 4 {d_{(i}}^{kl} d_{kl|j)} - 3 {d_{(i}}^{kl} d_{j)kl} + K K_{(ij)} - 2 K_{(i|k} {K^k}_{j)} ) + 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij}) $
$= - \alpha \delta^m_{(i} \delta^r_{j)} a_{m,r} + \alpha ( \gamma^{pq} \delta^m_{(i} \delta^r_{j)} - \gamma^{mr} \delta^p_{(i} \delta^q_{j)} ) d_{mpq,r} - 2 \alpha \delta^m_{(i} \delta^r_{j)} (\gamma^{kl} d_{klm})_{,r} + 2 \alpha \delta^m_{(i} \delta^r_{j)} \Gamma_{m,r} + \beta^k K_{(ij),k} + 2 K_{k(i} {\beta^k}_{,j)} + \alpha ( - a_{(i} a_{j)} + (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k - \Gamma_k) + 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kli} + 4 {d_{(i}}^{kl} d_{kl|j)} - 3 {d_{(i}}^{kl} d_{j)kl} + K K_{(ij)} - 2 K_{(i|k} {K^k}_{j)} ) + 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij}) $
$= - \alpha \delta^m_{(i} \delta^r_{j)} a_{m,r} + \alpha ( \gamma^{pq} \delta^m_{(i} \delta^r_{j)} - \gamma^{mr} \delta^p_{(i} \delta^q_{j)} - 2 \gamma^{mp} \delta^q_{(i} \delta^r_{j)} ) d_{mpq,r} + 2 \alpha \delta^m_{(i} \delta^r_{j)} \Gamma_{m,r} + \beta^k K_{(ij),k} + 2 K_{k(i} {\beta^k}_{,j)} + \alpha ( - a_{(i} a_{j)} + (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k - \Gamma_k) + 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kli} - 3 {d_{(i}}^{kl} d_{j)kl} + 8 {d_{(i}}^{kl} d_{kl|j)} + K K_{(ij)} - 2 K_{(i|k} {K^k}_{j)} ) + 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij}) $
matrix form
$\left[\matrix{ a_k \\ d_{kij} \\ K_{ij} \\ \Gamma_k }\right]_{,t} + \left[\matrix{ -\delta^m_k \beta^r & 0 & \alpha f \gamma^{pq} \delta^r_k & 0 \\ 0 & -\delta^m_k \delta^p_i \delta^q_j \beta^r & \alpha \delta^p_i \delta^q_j \delta^r_k & 0 \\ \alpha \delta^m_{(i} \delta^r_{j)} & \alpha ( \gamma^{(mr)} \delta^{(p}_{(i} \delta^{q)}_{j)} - \gamma^{(pq)} \delta^{(m}_{(i} \delta^{r)}_{j)} + 2 \gamma^{mp} \delta^q_{(i} \delta^r_{j)} ) & -\delta^p_i \delta^q_j \beta^r & -2 \alpha \delta^m_{(i} \delta^r_{j)} \\ 0 & 0 & \alpha (2 \delta^p_k \gamma^{qr} - \delta^r_k \gamma^{pq}) & -\delta^m_k \beta^r }\right] \left[\matrix{ a_m \\ d_{mpq} \\ K_{pq} \\ \Gamma_m }\right]_{,r} = \left[\matrix{ a_i {\beta^i}_{,k} + \alpha (a_k K (f - \alpha f') - 2 f {d_k}^{ij} K_{ij}) \\ d_{lij} {\beta^l}_{,k} + 2 d_{kl(i} {\beta^l}_{,j)} + \gamma_{l(i} {\beta^l}_{,j)k} - \alpha a_k K_{ij} \\ 2 K_{k(i} {\beta^k}_{,j)} + \alpha ( - a_{(i} a_{j)} + (2 {d_{(ij)}}^k - {d^k}_{(ij)}) (a_k - \Gamma_k) + 2 ({d^{kl}}_j - {d^{lk}}_j) d_{kli} - 3 {d_{(i}}^{kl} d_{j)kl} + 8 {d_{(i}}^{kl} d_{kl|j)} + K K_{(ij)} - 2 K_{(i|k} {K^k}_{j)} ) + 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij}) \\ - 2 ({d_{lk}}^r + {d^r}_{kl}) {\beta^l}_{,r} - (2 {d_n}^{nr} - {d^{rn}}_n) \gamma_{kl} {\beta^l}_{,r} + \gamma_{kl} \Gamma^r {\beta^l}_{,r} + \delta^r_k \Gamma_l {\beta^l}_{,r} + \gamma_{kl} {\beta^l}_{,ij} \gamma^{ij} - 2 \alpha K_{ik} \Gamma^i - 2 \alpha a^i K_{ki} + \alpha a_k K + 4 \alpha {d_i}^{ij} K_{kj} - 2 \alpha {d^{ji}}_i K_{kj} + 4 \alpha {d^{ij}}_k K_{ij} - 2 \alpha {d_k}^{ij} K_{ij} }\right]$