Is the matrix for the Killing form invertible for all semi-simple Lie algebras

I know that the Killing form is non-degenerate for semi-simple Lie Algebras. $$\Gamma(v,w) = \gamma_{ij} v^i w^j$$

I just wanted to double check that this was equivalent to saying that the Killing matrix $\gamma_{ij}$ was invertible for semi-simple Lie Algebras.

For any bilinear form (on a finite-dimensional vector space over a given field) it is true that the form is non-degenerate if and only if its representing matrix (w.r.t. any basis) is invertible. See e.g. Non-degenerate bilinear forms and invertible matrices.

By the way, I have not heard the nomenclature "the Killing matrix" before, and would like to point out that this representing matrix (although not its invertibility) depends on what basis we choose for the Lie algebra.