  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 $k$.
