I have to prove the following formula: $$\langle \nabla_X Y,Z\rangle=\frac{1}{2}(\partial_X\langle Y,Z\rangle+\partial_Y\langle Z,X\rangle-\partial_Z\langle X,Y\rangle+\langle Z,[X,Y]\rangle+\langle Y,[Z,X]\rangle-\langle X,[Y,Z]\rangle)$$
I start by computing the differential of the covariant derivative: \begin{align*} \partial_X \langle Y, Z \rangle &= \langle\nabla_X Y,Z\rangle + \langle Y,\nabla_X Z \rangle \\ \partial_Y \langle Z, X \rangle &= \langle\nabla_Y Z,X\rangle + \langle Z,\nabla_Y X \rangle \\ -\partial_Z \langle X, Y \rangle &= -\langle\nabla_Z X,Y\rangle - \langle X,\nabla_Z Y \rangle \end{align*} by adding up the above equations, we get \begin{align*} \partial_X\langle Y,Z\rangle + \partial_Y\langle Z,X\rangle - \partial_Z\langle X,Y \rangle &= \langle Y,\nabla_X Z-\nabla_Z X\rangle + \langle X,\nabla_Y Z-\nabla_Z Y \rangle + \langle Z,\nabla_X Y+ \nabla_Y X \rangle \end{align*} then, using the fact that the torsion is zero and considering $\nabla_X Y + \nabla_Y X = 2\nabla_X Y + \nabla_Y X - \nabla_X Y$ in the last right term, we get \begin{align*} \partial_X\langle Y,Z\rangle + \partial_Y\langle Z,X\rangle - \partial_Z\langle X,Y \rangle &=\langle Y,[X,Z]\rangle + \langle X,[Y,Z] \rangle - \langle Z,[X,Y] \rangle + 2\langle Z,\nabla_X Y \rangle \end{align*} Therefore \begin{align*} \langle \nabla_X Y, Z \rangle &= \dfrac{1}{2}\bigg( \partial_X\langle Y,Z\rangle + \partial_Y\langle Z,X\rangle - \partial_Z\langle X,Y \rangle - \langle X,[Y,Z] \rangle -\langle Y,[X,Z]\rangle + \langle Z,[X,Y] \rangle \bigg) \end{align*}
However, I get a different a different sign on the 5th element (i.e. $\langle Y,[X,Z]\rangle$) on the RHS. Can somebody suggest where I made a mistake?