# Riemannian metric and Laplacian coming from an invariant form on the lie algebra

Let $$G$$ be a semisimple real Lie group. Let $$\Delta \in U(\mathfrak g_{\mathbb C})$$ be the Casimir element associated to the Killing form on the complexified Lie algebra $$\mathfrak g_{\mathbb C}$$ of $$G$$.

On the Wikipedia article on the Casimir element, it says:

"the choice of an invariant bilinear form on $$\mathfrak g_{\mathbb C}$$ corresponds to a choice of bi-invariant Riemannian metric on $$G$$. Then under the identification of the universal enveloping algebra with the left invariant differential operators on $$G$$, the Casimir element maps to the Laplacian of $$G$$ (with respect to the corresponding bi-invariant metric)."

What is the Riemannian metric on $$G$$ associated to the Killing form? How is it defined? And what is meant by the Laplacian of $$G$$ with respect to a Riemannian metric? There seem to be several inequivalent notions of a Laplacian operator.