We consider the first two terms; the complete expression can then be obtained at any stage by interchanging $\nu$ and~$\sigma$ and subtracting. The additional terms introduced by the stars are by~\Eq{(86.2)} %[** TN: Re-breaking] \begin{multline*} -\frac{\dd}{\dd x_{\sigma}} (-g_{\mu}^{\epsilon} \kappa_{\nu} - g_{\nu}^{\epsilon} \kappa_{\mu} + g_{\mu\nu} \kappa^{\epsilon}) + (-g_{\mu}^{\alpha} \kappa_{\sigma} - g_{\sigma}^{\alpha} \kappa_{\mu} + g_{\mu\sigma} \kappa^{\alpha}) \{\alpha\nu, \epsilon\} \\ + (-g_{\alpha}^{\epsilon} \kappa_{\nu} - g_{\nu}^{\epsilon} \kappa_{\alpha} + g_{\alpha\nu} \kappa^{\epsilon}) \{\mu\sigma, \alpha\} \\ + (-g_{\mu}^{\alpha} \kappa_{\sigma} - g_{\sigma}^{\alpha} \kappa_{\mu} + g_{\mu\sigma} \kappa^{\alpha}) (-g_{\alpha}^{\epsilon} \kappa_{\nu} - g_{\nu}^{\epsilon} \kappa_{\alpha} + g_{\alpha\nu} \kappa^{\epsilon}) \\ = g_{\mu}^{\epsilon}\, \frac{\dd\kappa_{\nu}}{\dd x_{\sigma}} + g_{\nu}^{\epsilon}\, \frac{\dd\kappa_{\mu}}{\dd x_{\sigma}} - g_{\mu\nu}\, \frac{\dd\kappa^{\epsilon}}{\dd x_{\sigma}} - \frac{\dd g_{\mu\nu}}{\dd x_{\sigma}} \kappa^{\epsilon} \\ - \kappa_{\sigma} \{\mu\nu, \epsilon\} - \kappa_{\mu} \{\sigma\nu, \epsilon\} + g_{\mu\sigma} \{\alpha\nu, \epsilon\} \kappa^{\alpha} \\ - \kappa_{\nu} \{\mu\sigma, \epsilon\} - g_{\nu}^{\epsilon} \{\mu\sigma, \alpha\} \kappa_{\alpha} + \kappa^{\epsilon} [\mu\sigma, \nu] + g_{\mu}^{\epsilon} \kappa_{\sigma} \kappa_{\nu} + g_{\nu}^{\epsilon} \kappa_{\sigma} \kappa_{\mu} - g_{\mu\nu} \kappa_{\sigma} \kappa^{\epsilon} \\ + g_{\sigma}^{\epsilon} \kappa_{\mu} \kappa_{\nu} + g_{\nu}^{\epsilon} \kappa_{\mu} \kappa_{\sigma} - g_{\sigma\nu} \kappa_{\mu} \kappa^{\epsilon} - g_{\mu\sigma} \kappa^{\epsilon} \kappa_{\nu} - g_{\mu\sigma} g_{\nu}^{\epsilon} \kappa^{\alpha} \kappa_{\alpha} + g_{\mu\sigma} \kappa_{\nu} \kappa^{\epsilon}, \Tag{(87.2)} \end{multline*} which is equivalent to \[ g_{\mu}^{\epsilon}\, \frac{\dd\kappa_{\nu}}{\dd x_{\sigma}} + g_{\nu}^{\epsilon} (\kappa_{\mu})_{\sigma} - g_{\mu\nu} (\kappa^{\epsilon})_{\sigma} + g_{\nu}^{\epsilon} \kappa_{\mu} \kappa_{\sigma} - g_{\nu}^{\epsilon} g_{\mu\sigma} \kappa_{\alpha} \kappa^{\alpha} + g_{\mu\sigma} \kappa_{\nu} \kappa^{\epsilon}. \Tag{(87.3)} \] [To follow this reduction let the terms in~\Eq{(87.2)} be numbered in order from~$1$ to~$19$. It will be found that the following terms or pairs of terms are symmetrical in $\nu$ and~$\sigma$, and therefore disappear when the expression is completed, viz.\ $5$~and~$8$, $6$,~$11$, $12$ and $14$, $13$ and $17$,~$16$. Further $4$~and $10$ together give $-[\nu\sigma, \mu] \kappa^{\epsilon}$, which is rejected for the same reason. We combine $2$~and $9$ to give $g_{\nu}^{\epsilon} (\kappa_{\mu})_{\sigma}$. We exchange $7$ for its counterpart $-g_{\mu\nu} \{\alpha\sigma, \epsilon\} \kappa^{\alpha}$ in the remaining half of the expression, and combine it with~$3$ to give $-g_{\mu\nu} (\kappa^{\epsilon})_{\sigma}$.]