which lets us write out matrices for the adjoint action:
and from here it’s easy to calculate the Killing form. For example:
We can similarly calculate all the other values of the Killing form on basis elements.
So we can write down the matrix of :
And we can test this for degeneracy by taking its determinant to find . Since this is nonzero, we conclude that is nondegenerate, which we know means that is semisimple — at least in fields where .