Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. For $X\in \mathfrak{g}$, Let $\tilde{X}$ be the corresponding left invariant vector field on $G$.
I want to know $\nabla_{\tilde{X}} (\tilde{X})=0$ for all $X \in \mathfrak{g}$ implies $g(\tilde{Z}, [\tilde{X}, \tilde{Z}])=0$ for $X,Z\in \mathfrak{g}$, where $g(\cdot, \cdot)$ is a pseudo-Riemannian structure which satisfies \begin{align} 2g(X, \nabla_Z Y) = Z g(X,Y) + g(Z,[X,Y]) + Y g(X,Z) + g(Y,[X,Z]) - X g(Y,Z) - g(X,[Y,Z]) \end{align}
From above equation I tried to obtain $g(\tilde{Z}, [\tilde{X},\tilde{Z}])=0$. My first try is to put $Y=Z$, but it does not give desired results.
The question makes sense if $\nabla$ is the Levi-Civita connection of $g$. The conclusion seems to require also the hypothesis that $g$ is left-invariant, although this is not stated in the question. If this is assumed, expressions of the form $\tilde{X}g(\tilde{Y}, \tilde{Z})$ vanish because $g(\tilde{Y}, \tilde{Z})$ is constant, by left-invariance of $g$. In that case, $\nabla$ is torsion-free, so $[\tilde{X}, \tilde{Z}] = \nabla_{\tilde{X}}\tilde{Z} - \nabla_{\tilde{Z}}\tilde{X}$. Hence, using the definition of the Levi-Civita connection,
\begin{align*} \begin{split} g(\tilde{Z}, [\tilde{X}, \tilde{Z}])& = g(\tilde{Z}, \nabla_{\tilde{X}}\tilde{Z} - \nabla_{\tilde{Z}}\tilde{X})\\ & = \tfrac{1}{2}\tilde{X}g(\tilde{Z}, \tilde{Z}) + g(\nabla_{\tilde{Z}}\tilde{Z}, \tilde{X}) - \tilde{Z}g(\tilde{Z}, \tilde{X}) = g(\nabla_{\tilde{Z}}\tilde{Z}, \tilde{X}). \end{split} \end{align*} If you assume $\nabla_{\tilde{Z}}\tilde{Z} = 0$ for all $Z \in \mathfrak{g}$, the conclusion follows.