## Back to the Example

Let’s go back to our explicit example of and look at its Killing form. We first recall our usual basis:

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 .

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