A Magnus group construction for a class of Borcherds algebras

We construct a group associated to a class of Borcherds algebras that admit a direct sum decomposition into a Kac--Moody (or semi-simple) subalgebra and a pair of free Lie subalgebras. Such Borcherds algebras have no mutually orthogonal imaginary simple roots.Our group is a semi-direct product of a Kac--Moody (or semi-simple) group and a Magnus group of invertible formal power series corresponding to a basis of a certain highest weight module determined by the simple imaginary roots. We show that our group is independent of this choice of basis, up to isomorphism. We apply our construction to a number of concrete examples, such as certain Borcherds algebras formed using root lattices of hyperbolic Kac--Moody algebras, the Monster Lie algebra, Monstrous Lie algebras of Fricke type and the gnome Lie algebra.