The two following formulae express important properties of the Lagrangian function: \begin{align*} \mf{g}^{\mu\nu}\, \frac{\dd\mf{L}}{\dd\mf{g}^{\mu\nu}} &= -\mf{L}, \Tag{(58.71)} \\ \mf{g}_{\alpha}^{\mu\nu}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}} &= 2\mf{L}. \Tag{(58.72)} \end{align*} The first is obvious from~\Eq{(58.51)}. To prove the second, we have \begin{align*} \mf{g}_{\alpha}^{\mu\nu} &= \frac{\dd}{\dd x_{\alpha}} (g^{\mu\nu} \sqrt{-g}) = \sqrt{-g}\, \frac{\dd g^{\mu\nu}}{\dd x_{\alpha}} + g^{\mu\nu} \sqrt{-g}\, \{\alpha\epsilon, \epsilon\} \\ &= \sqrt{-g} \bigl[-\{\epsilon\alpha, \mu\} g^{\epsilon\nu} - \{\epsilon\alpha, \nu\} g^{\mu\epsilon} + \{\alpha\epsilon, \epsilon\} g^{\mu\nu}\bigr] \end{align*} by~\Eq{(30.1)} since the covariant derivative of~$g^{\mu\nu}$ vanishes. Hence by~\Eq{(58.52)} %[** TN: Re-breaking] \begin{multline*} \mf{g}_{\alpha}^{\mu\nu}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}} \\ = \sqrt{-g} \bigl[ \{\mu\nu, \alpha\} \{\epsilon\alpha, \mu\} g^{\epsilon\nu} + \{\mu\nu, \alpha\} \{\epsilon\alpha, \nu\} g^{\epsilon\mu} - \{\mu\nu, \alpha\} \{\alpha\epsilon, \epsilon\} g^{\mu\nu} \\ - \{\nu\beta, \beta\} g_{\mu}^{\alpha}\, \{\epsilon\alpha, \mu\} g^{\epsilon\nu} - \{\nu\beta, \beta\} g_{\mu}^{\alpha}\, \{\epsilon\alpha, \nu\} g^{\epsilon\mu} + \{\nu\beta, \beta\} g_{\mu}^{\alpha}\, \{\alpha\epsilon, \epsilon\} g^{\mu\nu}\bigr], \end{multline*} which by change of dummy suffixes becomes \begin{align*} &= \sqrt{-g} \bigl[ \{\beta\nu, \alpha\} \{\mu\alpha, \beta\} g^{\mu\nu} + \{\mu\beta, \alpha\} \{\nu\alpha, \beta\} g^{\nu\mu} - \{\mu\nu, \alpha\} \{\alpha\beta, \beta\} g^{\mu\nu} \\ &\qquad - \{\nu\beta, \beta\} \{\mu\alpha, \alpha\} g^{\mu\nu} - \{\alpha\beta, \beta\} \{\nu\mu, \alpha\} g^{\nu\mu} + \{\nu\beta, \beta\} \{\mu\epsilon, \epsilon\} g^{\mu\nu}\bigr] \\ &= 2\mf{L}\quad\text{by~\Eq{(58.1)}.} \end{align*} The equations \Eq{(58.71)} and \Eq{(58.72)} show that the Lagrangian function is a homogeneous function of degree~$-1$ in the ``coordinates'' and of degree~$2$ in the ``velocities.'' We can derive a useful expression for~$\mf{G}$ \begin{align*} \mf{G} &= \mf{g}^{\mu\nu} G_{\mu\nu} \\ &= \mf{g}^{\mu\nu}\, \frac{\dd}{\dd x_{\alpha}}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}} - \mf{g}^{\mu\nu}\, \frac{\dd\mf{L}}{\dd\mf{g}^{\mu\nu}}\quad\text{by~\Eq{(58.6)}} \\ &= \frac{\dd}{\dd x_{\alpha}} \left(\mf{g}^{\mu\nu}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}}\right) - \mf{g}_{\alpha}^{\mu\nu}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}} - \mf{g}^{\mu\nu}\, \frac{\dd\mf{L}}{\dd\mf{g}^{\mu\nu}} \\ &= \frac{\dd}{\dd x_{\alpha}} \left(\mf{g}^{\mu\nu}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}}\right) - \mf{L} \Tag{(58.8)} \end{align*} by \Eq{(58.71)} and~\Eq{(58.72)}.