A paper I'm reading in representation theory states the following result:

Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. Then there exists an upper triangular unipotent matrix $u$ such that $ux \space ^tu$ is diagonal.

I am looking for a reference for this result, or a suggestion on how to think about this to be able to prove it myself.

With respect to the standard basis in $F^n$, symmetric matrices correspond to symmetric bilinear forms, with invertible matrices corresponding to nondegenerate forms. I am trying to think about how to interpret the fact that the principal subdeterminants of $x$ are also nonzero, in terms of the symmetric form.