getSumDecomposition -- produces a simplified diagonal representative of a Grothendieck-Witt class or unstable Grothendieck-Witt class

Given a Grothendieck-Witt class or unstable Grothendieck-Witt class over $\mathbb{C},\mathbb{Q},\mathbb{R}$, or a finite field of characteristic not two, this method produces a simplified diagonal representative of the class. The result is a Grothendieck-Witt class or unstable Grothendieck-Witt class decomposed as a sum of some number of hyperbolic and rank one forms. In the unstable case, the result is the unstable Grothendieck-Witt class obtained by applying the method to the Grothendieck-Witt class factor of the unstable Grothendieck-Witt class.

We now describe the procedure for decomposing a Grothendieck-Witt class.

Over $\mathbb{C}$, symmetric bilinear forms are uniquely determined by their rank. Thus the decomposition of a form of rank $n$ is a sum of $\lfloor\frac{n}{2}\rfloor$ hyperbolic forms and a rank one form if $n$ is odd, or $\frac{n}{2}$ hyperbolic forms if $n$ is even.

Over $\mathbb{R}$, there are two square classes and a form is determined uniquely by its rank and signature [L05, II Proposition 3.5]. The form below is isomorphic to the form $\langle 1,-1,1\rangle$.

Over $\mathbb{Q}$, symmetric bilinear forms decompose into a sum of hyperbolic forms and its anisotropic part.

Over a finite field of characteristic not two, Grothendieck-Witt classes can similarly be diagonalized and decomposed.

[L05] Lam, T. Y., Introduction to Quadratic Forms over Fields, American Mathematical Society, 2005.