  We characterize H-like Lie algebras in terms of subspaces of cones over
conjugacy classes in $\mathfrak{so}(\mathbb{R}^q)$, translating the
classification problem for H-like Lie algebras to an equivalent problem in
linear algebra. We study properties of H-like Lie algebras, present new methods
for constructing them, including tensor products and central sums, and classify
H-like Lie algebras whose associated $J_Z$-maps have rank two for all nonzero
$Z$.
