  We show that the maximum cardinality of an equiangular line system in
$\mathbb R^{18}$ is at most $59$. Our proof includes a novel application of the
Jacobi identity for complementary subgraphs. In particular, we show that there
does not exist a graph whose adjacency matrix has characteristic polynomial
$(x-22)(x-2)^{42} (x+6)^{15} (x+8)^2$.
