Following Alcubierre "Introduction to 3+1 Numerical Relativity" section 5.4

primitive variables:
$\alpha =$ lapse
$\beta^i =$ shift
$\gamma_{ij} =$ spatial metric
$K_{ij} =$ extrinsic curvature
$K = tr K = {K^i}_i$
$R_{ij} =$ Ricci curvature of spatial hypersurface
$^4R_{ab} =$ Ricci curvature of spacetime

other variables:
$Q = $ slicing function
$Q = f(\alpha) K$ for Bona-Masso family of slicing (including "harmonic slicing" for when $f(\alpha) = 1$)
$a_i = (ln \alpha)_{,i}$
${b_i}^k = {\beta^k}_{,i}$
$d_{kij} = \frac{1}{2} \gamma_{ij,k}$
$\Gamma^k = \gamma^{ij} {\Gamma^k}_{ij}$
$= 2 {d_i}^{ik} - {d^{ki}}_{i}$
$V_k = {d_{km}}^m - {d^m}_{mk} = \frac{1}{2} ( {{d_i}^m}_m - \Gamma_i)$
${\Lambda^k}_{ij} = {d^k}_{ij} + \delta^k_{(i} ( a_{j)} + {d_{j)m}}^m - 2 {d^m}_{j)m})$
$ = {d^k}_{ij} + \delta^k_{(i} ( a_{j)} - {d_{j)m}}^m + 2 V_{j)})$
$ = {d^k}_{ij} + \delta^k_{(i} ( a_{j)} - \Gamma_{j)} )$
$\Lambda^k = \gamma^{ij} {\Lambda^k}_{ij}$
$P_k = {G^0}_k + a_m {K^m}_k - a_k K + {K^m}_n {d_{km}}^n - {K^m}_k {d_{ma}}^a - 2 K_{mn} {d^{mn}}_k + 2 K_{mk} {d_a}^{am}$
$\gamma = det || \gamma_{ij} ||$
$\phi = \frac{1}{12} ln \gamma$
$\Phi_i = \phi_{,i}$
$\tilde{d}_{ijk} = \frac{1}{2} \tilde{\gamma}_{jk,i}$
$\tilde{\gamma}_{ij} = e^{-4 \phi} \gamma_{ij}$, $\tilde{\gamma}^{ij} = e^{4 \phi} \gamma^{ij}$
${\tilde{\Gamma}^k}_{ij} = \frac{1}{2} \tilde{\gamma}^{kl} (\tilde{\gamma}_{li,j} + \tilde{\gamma}_{lj,i} - \tilde{\gamma}_{ij,l})$ $= {\Gamma^k}_{ij} - \frac{1}{3} (\delta^k_i {\Gamma^m}_{jm} + \delta^k_j {\Gamma^m}_{im} - \gamma_{ij} \gamma^{kl} {\Gamma^m}_{lm})$ $= {\Gamma^k}_{ij} - 2 (\delta^k_i \phi_{,j} + \delta^k_j \phi_{,i} - \gamma_{ij} \gamma^{kl} \phi_{,l})$
$\tilde{\Gamma}^i = \tilde{\gamma}^{jk} {\tilde{\Gamma}^i}_{jk} = -{\tilde{\gamma}^{ij}}_{,j} = e^{4\phi} \Gamma^i + 2 \tilde{\gamma}^{ij} \phi_{,j}$
$A_{ij} = K_{ij} - \frac{1}{3} \gamma_{ij} K$
$\tilde{A}_{ij} = e^{-4 \phi} A_{ij} $
${\tilde{\Lambda}^k}_{ij} = ( {\tilde{d}^k}_{ij} + \delta^k_{(i} ( a_{j)} - \tilde{\Gamma}_{j)} + 2 \Phi_{j)} ) )^{TF}$

Projections of Einstein Field Equations:
$H = R + K^2 - K_{ij} K^{ij} - 16 \pi \rho = 0$ is the Hamiltonian constraint.
$M^i = D_j (K^{ij} - \gamma^{ij} K) - 8 \pi j^i = 0$ is the momentum constraint.

stress-energy constraints (eqns 2.5.10, 2.5.11, 2.5.12):
$0 = H = n^a n^b G_{ab} - 8 \pi \rho$
$0 = M_u = -n^a {\gamma^b}_u G_{ab} - 8 \pi j_u$
$0 = E_{uv} = {\gamma^a}_u {\gamma^b}_v G_{ab} - 8 \pi S_{uv}$

stress-energy variable definitions (derived from above, also 2.4.12):
$\rho = n^a n^b T_{ab}$
$j_u = -n^a {\gamma^b}_u T_{ab}$, B&S uses $S_u = -j_u$.
$S_{uv} = {\gamma^a}_u {\gamma^b}_v T_{ab}$

stress-energy variable definitions:
$16 \pi \rho = R + K^2 - K_{ij} K^{ij}$ (A 2.4.10, B&S 2.90)
$8 \pi j_u = D_j (K^{ij} - \gamma^{ij} K)$ (A 2.4.11, B&S 2.96
$S_{ij} = -^4R_{ij} + K K_{ij} - 2 K_{ij} {K^k}_j + 4 d_{kmi} {d^{km}}_j + {\Gamma^k}_{km} {\Gamma^m}_{ij} - \Gamma_{ikm} {\Gamma_j}^{km} + 2 (a^k - 2 {d_m}^{km}) d_{(ij)k} + a_{(i} (2 V_{j)} - {d_{j)k}}^k)$


"equal up to principle part" approximations (neglecting non-derivatives):
$H \approx R \approx -2 {V^m}_{,m}$ up to principle part
$M_i \approx 2{K^m}_{[i,m]}$ up to principle part

operators:
$D_i = $ spatially projected covariant derivative
$\mathcal{L} =$ Lie derivative
$\partial_0 = \partial_t - \beta^k \partial_k$ is the derivative in the time direction "up to principle part" or something like that
$\frac{d}{dt} = \partial_t - \mathcal{L}_\vec\beta$ is the derivative in the time direction

eigenfield meaning:
for state vector $u_a = (a_i, d_{ijk}, K_{ij})$,
eigenfields are of the form $w_a = C_{ab} u_b$
that evolve "up to principle part" with $\partial_t w_a + \lambda \partial_x w_a = 0$
for some associated eigenvalue $\lambda$