Orthogonal basis for bilinear forms I found the following claim all over the physics literature and online:

Let $B$ be a non-degenerate symmetric bilinear form on a finite-dimensional $\mathbb K$-vector space with $\mathbb K$ a field of characteristic $\ne 2$. Then $B$ has an orthogonal basis.

(The specific context in the physics books is when choosing an orthogonal basis for the Killing form of a semisimple Lie algebra (and sometimes they also assume the Lie algebra to be compact, I do not know why)).
Where can I find a proof of this claim?
Related: This post

The proof basically consists of repeated applications of completing the square (the method sometimes referred to as Lagrange reduction), and can be found here.
