The PBW conjecture for mock-Lie algebras

Let $L$ be a mock-Lie algebra, and let $U(L)$ denote its universal enveloping algebra. The Poincaré–Birkhoff–Witt theorem holds for $L$ when the ordered monomials in a basis of $L$ form a basis of $U(L)$. The PBW conjecture for mock-Lie algebras. The mock-Lie algebras for which the Poincaré–Birkhoff–Witt theorem holds are exactly the abelian algebras. The PBW theorem is known to fail generally for mock-Lie algebras because the defining relations of $U(L)$ need not be closed under compositions; the conjecture identifies the abelian algebras as precisely the exceptional cases.