Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms

In this paper we prove a correspondence between a canonical degree six covariant of binary quartic forms $F$ and a cubic covariant of a pair of ternary quadratic forms $(f_A, f_B)$. In the process we obtain a canonical way to diagonalize a pair of $n$-ary quadratic forms over any field $K$ of characteristic zero. As a corollary, we give a precise criterion to decide whether a pair of $n$-ary quadratic forms over $\mathbb{Q}$ is diagonalizable over $\mathbb{Q}$.