HilbertSymbolReal(a,b,p)
The HasseWitt invariant of a diagonal form $\langle a_1,\ldots,a_n\rangle$ over a field $K$ is defined to be the product $\prod_{i<j} \phi(a_i,a_j)$ where $\phi \colon K \times K \to \left\{\pm 1\right\}$ is any symbol (see e.g. [MH73, III.5.4] for a definition). It is a classical result of Hilbert that over a local field of characteristic not equal to two, there is one and only symbol, $(,)_p$ called the Hilbert symbol ([S73, Chapter III]) computed as follows:
$(a,b)_{\mathbb{R}} = \begin{cases} 1 & z^2 = ax^2 + by^2 \text{ has a nonzero solution in } {\mathbb{R}}^3 \\ 1 & \text{otherwise.} \end{cases}$
$(a,b)_{\mathbb{R}}$ will equal 1 unless both $a,\,b$ are negative.
Consider the example, that $z^2=3x^22y^2/3$ does not admit a nonzero solution. Thus:

Computing HasseWitt invariants is a key step in classifying symmetric bilinear forms over the rational numbers, and in particular certifying their (an)isotropy.
Citations:
The object HilbertSymbolReal is a method function.