An arithmetic count of the lines meeting four lines in P^3

We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field $k$, this enrichment counts the number of lines meeting four lines defined over $k$ in $\mathbb{P}^3_k$, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in $\mathbb{A}^1$-homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms. In the appendix, the condition that the four lines each be defined over $k$ is relaxed to the condition that the set of four lines being defined over $k$.