I've come across an odd-looking expression and oh how I wish I had a more elegant description of it. Maybe someone who enjoys symmetric bilinear forms in characteristic two will recognize it? Or someone who chooses bases and recognizes sums with indices instantly?

Here's the setup: $X$ is a finite-rank free ${\mathbb Z}$-module, of rank $r$, with basis $\{ x_1, \ldots, x_r \}$. $g \in GL_r({\mathbb Z})$ is a matrix (change of basis matrix). Finally, $C \in (X \otimes X) \otimes {\mathbb Z / 2 \mathbb Z}$ is a symmetric tensor, modulo $2$. Concretely, with respect to the basis, $C = (c_{ij})$ with $c_{ij} \in {\mathbb Z / 2 \mathbb Z}$ and $c_{ij} = c_{ji}$ for all $1 \leq i,j \leq r$.

From $g$ and $C$, I am encountering the vector $a = \sum_j a_j x_j \in X \otimes {\mathbb Z / 2 \mathbb Z}$, where $$a_j = \left( \sum_{i=1}^r c_{ii} \frac{ g_{ij} (g_{ij} - 1)}{2} \right) + \sum_{k < \ell} c_{k \ell} g_{k j} g_{\ell j}.$$

I'm encountering this vector in two very specific settings related to metaplectic groups. It looks like this vector $a$ should come from something "natural" in linear algebra or combinatorics.

Has anyone seen a vector like $a$ before? Contexts?