  We consider integrable category $\mathcal{O}$ representations of
Borcherds--Kac--Moody algebras whose Cartan matrix is finite dimensional, and
determine the necessary and sufficient conditions for which the tensor product
of irreducible representations from this category is isomorphic to another.
This result generalizes a fundamental result of C. S. Rajan on unique
factorization of tensor products of finite dimensional irreducible
representations of finite dimensional simple Lie algebras over complex numbers.
