The suffix~$\epsilon$ may be lowered. Thus \begin{align*} B_{\mu\nu\sigma\rho} &= g_{\rho\epsilon} B_{\mu\nu\sigma}^{\epsilon} \\ &= \{\mu\sigma, \alpha\} [\alpha\nu, \rho] - \{\mu\nu, \alpha\} [\alpha\sigma, \rho] - \frac{\dd}{\dd x_{\nu}} [\mu\sigma, \rho] - \frac{\dd}{\dd x_{\sigma}} [\mu\nu, \rho] \\ &\qquad- \{\mu\sigma, \alpha\}\, \frac{\dd g_{\rho\alpha}}{\dd x_{\nu}} - \{\mu\nu, \alpha\}\, \frac{\dd g_{\rho\alpha}}{\dd x_{\sigma}}, \Tag{(34.45)} \intertext{where $\epsilon$~has been replaced by~$\alpha$ in the last two terms,} &= -\{\mu\sigma, \alpha\} [\rho\nu, \alpha] + \{\mu\nu, \alpha\} [\rho\sigma, \alpha] \\ &\qquad+\tfrac{1}{2} \left( \frac{\dd^{2} g_{\rho\sigma}}{\dd x_{\mu}\, \dd x_{\nu}} + \frac{\dd^{2} g_{\mu\nu}}{\dd x_{\rho}\, \dd x_{\sigma}} - \frac{\dd^{2} g_{\mu\sigma}}{\dd x_{\rho}\, \dd x_{\nu}} - \frac{\dd^{2} g_{\rho\nu}}{\dd^{2} x_{\mu}\, \dd x_{\sigma}}\right), \Tag{(34.5)} \end{align*} by \Eq{(27.5)} and~\Eq{(27.1)}. It will be seen from~\Eq{(34.5)} that $B_{\mu\nu\sigma\rho}$, besides being antisymmetrical in~$\nu$ and~$\sigma$, is also antisymmetrical in $\mu$ and~$\rho$. Also it is symmetrical for the double interchange $\mu$ and~$\nu$, $\rho$ and~$\sigma$. It has the further cyclic property \[ B_{\mu\nu\sigma\rho} + B_{\mu\sigma\rho\nu} + B_{\mu\rho\nu\sigma} = 0, \Tag{(34.6)} \] as is easily verified from~\Eq{(34.5)}. The general tensor of the fourth rank has $256$~different components. Here the double antisymmetry reduces the number (apart from differences of sign) to $6 \times 6$. $30$~of these are paired because $\mu$,~$\rho$ can be interchanged with $\nu$,~$\sigma$; but the remaining $6$~components, in which $\mu$,~$\rho$ is the same pair of numbers as $\nu$,~$\sigma$, are without partners. This leaves $21$~different components, between which \Eq{(34.6)}~gives only one further relation. We conclude that the Riemann-Christoffel tensor has $20$~\emph{independent} components\footnotemark.\footnotetext {Writing the suffixes in the order $\mu\rho\sigma\nu$ the following scheme gives $21$~different components: \[ \begin{array}{*{7}{c}} 1212 & 1223 & 1313 & 1324 & 1423 & 2323 & 2424 \\ 1213 & 1224 & 1314 & 1334 & 1424 & 2324 & 2434 \\ 1214 & 1234 & 1323 & 1414 & 1434 & 2334 & 3434 \\ \end{array} \] with the relation $1234 - 1324 + 1423 = 0$. If we omit those containing the suffix~$4$, we are left with $6$~components in three-dimensional space. In two dimensions there is only the one component~$1212$.}