Simultaneous diagonalization of two rational forms It is known that any two real quadratic forms are equivalent iff they have the same signature. If we consider rational quadratic forms, they are $\mathbb{Q}$-equivalent iff the have the same signature, discriminant and Hasse invariants.
For the real case it is also known that if we have two forms $F$ and $G$ where $G$ is positive-definite then we can simultaneously diagonalize them so that $G$ will become identity form.
Is there some analogue of simultaneous diagonalization of two rational forms?
 A: There is a complete theory to determine when two quadratic forms defined
over a field of characteristic not 2 can be simultaneously diagonalized.
The theory can be easily applied to the rational numbers.  The complete answer is not easy to state because it involves a number of cases.  Although the theory goes back to L.E. Dickson and often appeared in advanced algebra texts during the first half of the 20th century, the theory does not appear as often now.
The first good modern treatment I know was published by Waterhouse.  See
Waterhouse, William C. Pairs of quadratic forms. Invent. Math. 37 (1976), no. 2, 157–164.
The complete story appears there, but many details of the proof are left to the reader.
A very complete treatment with all details given (and possibly a bit too tedious) appears in the following paper.
Leep, David B.; Schueller, Laura Mann Classification of pairs of symmetric and alternating bilinear forms. Exposition. Math. 17 (1999), no. 5, 385–414.
In both papers, the basic idea is to find a set of invariants that classifies pairs of quadratic forms up to isometry.  Along the way, one finds a canonical form for each isometry class of pairs. The theory allows one to determine when the canonical form for an isometry class of pairs turns out to be a simultaneously diagonalized pair.
