Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. I know that if $g$ is semi-simple then the Laplace-Beltrami operator on $G$ agrees with the Casimir element and therefore commutes with with the elements of $\mathfrak{g}$. Here we use the Riemmanian metric given by the Killing form.

In the general case, we can write $\mathfrak{g} = \mathfrak g' \oplus \mathfrak g''$ with $\mathfrak g'$ semi-simple. Now we can endow $G$ with a Riemmaniam metric given by the Killing form on $\mathfrak g'$ and any other metric on $\mathfrak g''$.

Is it possible to show that in this case the Laplace-Beltrami operator still commutes with the elements of $\mathfrak g$?

If this is a well known result, what is a good reference for this?