The structure of non-Fricke Monstrous Lie algebras

We consider Borcherds algebras with no real roots and the property that all zeroes in the Borcherds Cartan matrix occur in a single diagonal zero block. It follows that all other entries of the matrix are negative. We give a structure theorem for these Borcherds algebras, decomposing them into free, Heisenberg and abelian subalgebras. We show that a class of such Borcherds algebras are the Monstrous Lie algebras associated to non-Fricke elements of the Monster finite simple group. This new perspective on their structure gives an efficient method to compute their twisted denominator formulas.