Proving an identity of Riemannian tensor

Question

starting by this definition of the Riemann tensor : let $X$,$Y$ and $Z$ be vector fields $$ R(X,Y)Z= \nabla_X\nabla_Y Z-\nabla_Y\nabla_X Z-\nabla_{[X,Y]}Z $$ I tried to proof this identity from Wikipedia using this difenition : $$ {\langle R(u,v)w,z\rangle =-\langle R(u,v)z,w\rangle } $$ Here the bracket $ \langle\, ,\rangle $ refers to the inner product on the tangent space induced by the metric tensor.

Answer 1

We show that $\langle R(X,Y)Z,\,Z\rangle=0$, which entails your result.

For this, notice that $$\langle \nabla_X \nabla_Y Z,\,Z\rangle=\mathscr{L}_X\langle \nabla_Y Z,\, Z\rangle -\langle \nabla_X Z,\,\nabla_Y Z\rangle,$$ and same for the second term.

Thus it remains to prove that $$\mathscr{L}_X\langle \nabla_Y Z,\,Z\rangle-\mathscr{L}_Y\langle \nabla_X Z,\,Z \rangle=\langle \nabla_[X,Y] Z\,Z \rangle.$$

Multiplying by two, the equality is equivalent to $$\mathscr{L}_X\mathscr{L}_Y \langle Z,\,Z \rangle-\mathscr{L}_Y\mathscr{L}_X\langle Z,\,Z \rangle=\mathscr{L}_{[X,Y]}\langle Z,\, Z\rangle,$$ which is well-known to be true.