Appendix: Mathematical Details

### Divergence of the Einstein Tensor

We verify here that the divergence of the Einstein vector-valued 1-form $\GG$ vanishes, that is, that \begin{equation} d\gamma^i + \omega^i{}_j\wedge\gamma^j = 0 \label{diveineq} \end{equation} where \begin{equation} \gamma^i = \frac12 \epsilon_\ell{}^{ijk} \,\Omega_{jk} \wedge \sigma^\ell \end{equation} as shown in §Components of the Einstein Tensor. Using the Bianchi identities and the structure equations to evaluate the first term, we have \begin{align} 2 &(d\gamma^i + \omega^i{}_j\wedge\gamma^j) \nonumber\\ =& \epsilon_l{}^{ijk} \left( -\omega_{jm}\wedge\Omega^m{}_k\wedge\sigma^l +\Omega_{jm}\wedge\omega^m{}_k\wedge\sigma^l -\Omega_{jk}\wedge\omega^l{}_m\wedge\sigma^m \right) \nonumber\\ &\qquad +\epsilon_l{}^{mjk} \omega^i{}_m\wedge\Omega_{jk}\wedge\sigma^l \label{dgamma0} \end{align}

Consider first the very last term of (\ref{dgamma0}). The antisymmetry of the Levi-Civita symbol implies that $l$, $m$, $j$, $k$ must be distinct, so exactly one of them must be equal to $i$. The antisymmetry of the connection 1-forms now implies (in an orthonormal frame) that $i\ne m$. We have therefore shown that \begin{align} \epsilon_l{}^{mjk} \omega^i{}_m\wedge\Omega_{jk}\wedge\sigma^l &= \epsilon_l{}^{mik} \omega^i{}_m\wedge\Omega_{ik}\wedge\sigma^l + \epsilon_l{}^{mji} \omega^i{}_m\wedge\Omega_{ji}\wedge\sigma^l \nonumber\\ &\qquad + \epsilon_i{}^{mjk} \omega^i{}_m\wedge\Omega_{jk}\wedge\sigma^i \nonumber\\ &= 2\epsilon_l{}^{mji} \omega^i{}_m\wedge\Omega_{ji}\wedge\sigma^l + \epsilon_i{}^{mjk} \omega^i{}_m\wedge\Omega_{jk}\wedge\sigma^i \label{dgamma1} \end{align} (in an orthonormal frame), where there is no sum over $i$. Applying similar reasoning to the first two terms on the right-hand side of (\ref{dgamma0}), we see that $m$ can only take on the values $i$ and $l$ (in an orthonormal frame). Thus, \begin{align} \epsilon_l{}^{ijk} & \left( -\omega_{jm}\wedge\Omega^m{}_k\wedge\sigma^l +\Omega_{jm}\wedge\omega^m{}_k\wedge\sigma^l \right) \nonumber\\ &= 2\epsilon_l{}^{ijk} \Omega_{jm}\wedge\omega^m{}_k\wedge\sigma^l \nonumber\\ &= 2\epsilon_l{}^{ijk} \Omega_{ji}\wedge\omega^i{}_k\wedge\sigma^l + 2\epsilon_l{}^{ijk} \Omega_{jl}\wedge\omega^l{}_k\wedge\sigma^l \end{align} (in an orthonormal frame), where there is a sum over $l$ (and of course also over $j$ and $k$), but not over $i$. Relabeling indices and reordering factors brings this to the form \begin{align} \epsilon_l{}^{ijk} &\left( -\omega_{jm}\wedge\Omega^m{}_k\wedge\sigma^l +\Omega_{jm}\wedge\omega^m{}_k\wedge\sigma^l \right) \nonumber\\ &= 2\epsilon_l{}^{ijm} \omega^i{}_m\wedge\Omega_{ji}\wedge\sigma^l + 2\epsilon_l{}^{ijk} \Omega_{jl}\wedge\omega^l{}_k\wedge\sigma^l \label{dgamma2} \end{align} Finally, in the remaining term of (\ref{dgamma0}), $m$ can be $i$, $j$, or $k$, but not $l$ (in an orthonormal frame), leading to \begin{align} -\epsilon_l{}^{ijk} \Omega_{jk}\wedge\omega^l{}_m\wedge\sigma^m &= -\epsilon_l{}^{ijk} \Omega_{jk}\wedge\omega^l{}_i\wedge\sigma^i - \epsilon_l{}^{ijk} \Omega_{jk}\wedge\omega^l{}_j\wedge\sigma^j \nonumber\\ &\qquad - \epsilon_l{}^{ijk} \Omega_{jk}\wedge\omega^l{}_k\wedge\sigma^k \nonumber\\ &= -\epsilon_l{}^{ijk} \Omega_{jk}\wedge\omega^l{}_i\wedge\sigma^i - 2\epsilon_l{}^{ijk} \Omega_{jk}\wedge\omega^l{}_j\wedge\sigma^j \nonumber\\ &= -\epsilon_l{}^{ijk} \Omega_{jk}\wedge\omega^l{}_i\wedge\sigma^i - 2\epsilon_l{}^{ijk} \Omega_{jk}\wedge\omega^l{}_j\wedge\sigma^j \end{align} where there is a sum over $j$ and $k$ (and $l$), but not $i$. Relabeling indices and reordering factors leads to \begin{equation} -\epsilon_l{}^{ijk} \Omega_{jk}\wedge\omega^l{}_m\wedge\sigma^m = \epsilon_m{}^{ijk} \omega^m{}_i\wedge\Omega_{jk}\wedge\sigma^i - 2\epsilon_k{}^{ilj} \Omega_{lj}\wedge\omega^k{}_l\wedge\sigma^l \label{dgamma3} \end{equation} If we now add up (\ref{dgamma1}), (\ref{dgamma2}), and (\ref{dgamma3}), and carefully raise and lower (and reorder) some indices, everything cancels, and we have verified that (\ref{diveineq}) holds in an orthonormal basis. A similar computation, using the general expression of metric compatibility, establishes the same result in an arbitrary basis.

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