Faraday tensor defined using vectors:

From Baumgarate & Shapiro, eqn 5.114:
$F^{ab} = \frac{1}{c} n^a E^b - \frac{1}{c} n^b E^a + n_d \epsilon^{dabc} B_c$
$E^a = c F^{ab} n_b$
$B^a = \frac{1}{2} \epsilon^{abcd} n_b F_{dc}$
This fits our convention above, which means if I were to negative the E components of F (like most texts do) then I would have to negative n, or the definition of E wrt F and n, as well.

For ADM $n_a = (-\alpha, 0)$, $n^a = (1/\alpha, -\beta^i/\alpha)$, where the first component is time, this means:

$F^{0i} = \frac{1}{c} n^0 E^i - \frac{1}{c} n^i E^0 + n_d \epsilon^{d0ic} B_c = \frac{1}{c} \frac{1}{\alpha} E^i$
$E^a = c F^{ab} n_b = -\alpha c F^{a0} = \alpha c F^{0a}$

Since $F^{ab}$ is antisymmetric, this means $F^{tt} = 0$, and $E^t = 0$, so $E^a n_a = 0$, so $E^a$ is a spatial vector.
$E_a = g_{ab} E^b = g_{aj} E^j = \pmatrix{ \beta_j E^j & \gamma_{ij} E^j } = \pmatrix{\beta^j E_j & E_i}$
so $E_i = g_{ia} E^a = g_{ij} E^j$ and $E_0 = \beta^j E_j$

Now $n_d \epsilon^{dabc} = n_0 \epsilon^{0abc} = -\alpha \epsilon^{0abc} = \epsilon^{abc}$ where the last term is purely spatial (being orthogonal to $t$), so any timelike components of $abc$ are cancelled.
$F^{ij} = n_d \epsilon^{dijc} B_c = -\alpha \epsilon^{0ijk} B_k = \epsilon^{ijk} B_k$

$F^{ab} = \downarrow a(i) \overset{\rightarrow b(j)}{\pmatrix{ 0 & \frac{1}{c} \frac{1}{\alpha} E^j \\ -\frac{1}{c} \frac{1}{\alpha} E^i & \epsilon^{ijk} B_k }}$

${F^a}_b = F^{au} g_{ub} = \pmatrix{ 0 & \frac{1}{c} \frac{1}{\alpha} E^k \\ -\frac{1}{c} \frac{1}{\alpha} E^i & \epsilon^{ikl} B_l } \pmatrix{-\alpha^2 + \beta^2 & \beta_j \\ \beta_k & \gamma_{kj}}$
${F^a}_b = \pmatrix{ \frac{1}{c} \frac{1}{\alpha} E^k \beta_k & \frac{1}{c} \frac{1}{\alpha} E_j \\ \frac{1}{c} (\alpha - \frac{1}{\alpha} \beta^2) E^i + {\epsilon^i}_{kl} \beta^k B^l & -\frac{1}{c} \frac{1}{\alpha} E^i \beta_j + {\epsilon^i}_{jk} B^k }$

$F_{ab} = g_{au} {F^u}_b = \pmatrix{-\alpha^2 + \beta^2 & \beta_k \\ \beta_i & \gamma_{ik}} \pmatrix{ \frac{1}{c} \frac{1}{\alpha} E^l \beta_l & \frac{1}{c} \frac{1}{\alpha} E_j \\ \frac{1}{c} (\alpha - \frac{1}{\alpha} \beta^2) E^k + {\epsilon^k}_{lm} \beta^l B^m & -\frac{1}{c} \frac{1}{\alpha} E^k \beta_j + {\epsilon^k}_{jl} B^l } $
$F_{ab} = {F_a}^u g_{ub} = \pmatrix{ -\frac{1}{c} \frac{1}{\alpha} \beta_k E^k & \frac{1}{c} \frac{1}{\alpha} (-\alpha^2 + \beta^2) E^l - \alpha \beta_k \epsilon^{0klm} B_m \\ -\frac{1}{c} \frac{1}{\alpha} E_i & \frac{1}{c} \frac{1}{\alpha} \beta_i E^l - \alpha \gamma_{ik} \epsilon^{0klm} B_m } \pmatrix{ -\alpha^2 + \beta^2 & \beta_j \\ \beta_l & \gamma_{lj} }$
$F_{ab} = \pmatrix{ \frac{1}{c} (-\alpha^2 + \beta^2) (-\frac{1}{\alpha} \beta_k E^k) + \beta_l (\frac{1}{c} \frac{1}{\alpha} (-\alpha^2 + \beta^2) E^l + \beta_k \epsilon^{klm} B_m) & \frac{1}{c} \beta_j (-\frac{1}{\alpha} \beta_k E^k) + \gamma_{lj} (\frac{1}{c} \frac{1}{\alpha} (-\alpha^2 + \beta^2) E^l + \beta_k \epsilon^{klm} B_m) \\ \frac{1}{c} (-\alpha^2 + \beta^2) (-\frac{1}{\alpha} E_i) + \beta_l (\frac{1}{c} \frac{1}{\alpha} \beta_i E^l + \gamma_{ik} \epsilon^{klm} B_m) & \beta_j (-\frac{1}{c} \frac{1}{\alpha} E_i) + \gamma_{lj} (\frac{1}{c} \frac{1}{\alpha} \beta_i E^l + \gamma_{ik} \epsilon^{klm} B_m) }$
$F_{ab} = \pmatrix{ 0 & - ( \frac{1}{c} \alpha E_j - \frac{1}{c} \frac{1}{\alpha} \beta^2 E_j + \frac{1}{c} \frac{1}{\alpha} \beta_j \beta^k E_k + \gamma_{kj} \beta_l \epsilon^{klm} B_m ) \\ \frac{1}{c} \alpha E_i - \frac{1}{c} \frac{1}{\alpha} \beta^2 E_i + \frac{1}{c} \frac{1}{\alpha} \beta_i \beta^k E_k + \gamma_{ik} \beta_l \epsilon^{klm} B_m & \frac{1}{c} \frac{1}{\alpha} \beta_i E_j - \frac{1}{c} \frac{1}{\alpha} \beta_j E_i + \gamma_{ik} \gamma_{jl} \epsilon^{klm} B_m }$
$F_{ab} = \pmatrix{ 0 & - ( \frac{1}{c} \alpha E_j + 2 \frac{1}{c} \frac{1}{\alpha} \beta^k \beta_{[j} E_{k]} + \epsilon_{jkl} \beta^k B^l ) \\ \frac{1}{c} \alpha E_i + 2 \frac{1}{c} \frac{1}{\alpha} \beta^k \beta_{[i} E_{k]} + \epsilon_{ikl} \beta^k B^l & 2 \frac{1}{c} \frac{1}{\alpha} \beta_{[i} E_{j]} + \epsilon_{ijk} B^k }$