I'm rewriting O'Neill's "Elementary Differential Geometry"'s section on connection forms in Lorentz-Minkowski space $\Bbb L^3$, and I'm having trouble proving the second structural equations $${\rm d}\omega_{ij} = \sum_k \omega_{ik} \wedge \omega_{\color{blue}{kj}}.$$ I'm using the signature $(+,+,-)$, that is, $$G = \begin{pmatrix} 1& 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{pmatrix}.$$ I know for a fact that this result is true, I've seen it in a more general form in his other book, "The Geometry of Kerr Black Holes". The proof there is overkill for what I'm doing here. Since these sign conventions are such a pain, I'll put the definitions I'm working with here, along with two useful results that I've proved:
Definition: Given a frame field $\{{\bf E}_i\}_{i = 1}^3$, the $1$-forms defined by $$\omega_{ij}({\bf V}) = \epsilon_j \langle \nabla_{\bf V}{\bf E}_i, {\bf E}_j \rangle$$ are called the connection forms of the given frame field. Also, $\omega_{ij} = -\epsilon_i \epsilon_j \omega_{ji}$, where $\epsilon_i = \langle {\bf E}_i,{\bf E}_i\rangle$.
Definition: Given a matrix $A = (a_{ij})_{1 \leq i,j \leq 3}$, where the $a_{ij}$ are functions, we define ${\rm d}A = ({\rm d}a_{ij})_{1 \leq i,j \leq 3}$ and $A_\epsilon = (\epsilon_i a_{ij})_{1 \leq i,j \leq 3}$.
Theorem: Let $\{{\bf E}_i\}_{i = 1}^3$ be a frame field, and $A = (a_{ij})_{1 \leq i,j\leq 3}$ be its attitude matrix, that is, ${\bf E}_i = \sum_j a_{ij} \partial_j$. If $\omega = (\omega_{ij})_{1 \leq i,j \leq 3}$, then $$\omega = {\rm d}A \cdot GA_\epsilon^{ \ t},$$ where the product in the right side is to be seen as the wedge product. Note: $A_\epsilon^{ \ t} \neq A_{\ \epsilon}^t$.
Lemma: With notation as above: $$A \cdot GA_{\epsilon}^{\ t} = {\rm Id_3}.$$
With this I can go on. I have: $$\begin{align}\omega &= {\rm d}A \cdot GA_\epsilon^{ \ t} \\ &\implies {\rm d}\omega = {\rm d}( {\rm d}A \cdot GA_\epsilon^{ \ t}) \\ &\implies {\rm d}\omega = - {\rm d}A\cdot {\rm d}(GA_\epsilon^{\ t}) \\ &\implies {\rm d}\omega = - {\rm d}A \cdot G ({\rm d}A)_\epsilon^{\ t} \\ &\implies{\rm d}\omega = -{\rm d}A \cdot GA_{\epsilon}^{\ t} \cdot A \cdot G^t({\rm d}A)_\epsilon^{\ t} \\ &\implies {\rm d}\omega = -\omega \cdot ({\rm d}A)_\epsilon G A^t\end{align}.$$
Since I don't know how to go from here, I'll open both sides. Brace yourself, it'll be ugly.
$$\begin{align} {\rm d}\omega_{ij} &= -\sum_k \omega_{ik} \wedge (({\rm d}A)_\epsilon G A^t)_{kj}\\ &= -\sum_k \omega_{ik} \wedge \left(\sum_l \epsilon_k {\rm d}a_{kl}(GA^t)_{lj}\right) \\ &= -\sum_k \epsilon_k\omega_{ik} \wedge \left(\sum_l {\rm d}a_{kl} (g_{l1}a_{j1} + g_{l2}a_{j2}+g_{l3}a_{j3})\right) \\ &= -\sum_k \epsilon_k \omega_{ik } \wedge ({\rm d}a_{k1}a_{j1}+{\rm d}a_{k2}a_{j2} - {\rm d}a_{k3}a_{j3}) \\ &= \sum_{k} \omega_{ik} \wedge (-\epsilon_k \epsilon_j \omega_{kj}) \\ &= \sum_k \omega_{ik} \wedge \omega_{\color{red}{jk}}\end{align}.$$
The indices are swapped in this last $\omega$ and I can't find the mistake. Please help.