  We show that Automorphic Lie Algebras which contain a Cartan subalgebra with
a constant spectrum, called hereditary, are completely described by 2-cocycles
on a classical root system taking only two different values. This observation
suggests a novel approach to their classification. By determining the values of
the cocycles on opposite roots, we obtain the Killing form and the
abelianisation of the Automorphic Lie Algebra. The results are obtained by
studying equivariant vectors on the projective line. As a byproduct, we
describe a method to reduce the computation of the infinite dimensional space
of said equivariant vectors to a finite dimensional linear computation and the
determination of the ring of automorphic functions on the projective line.
