Generalized spin representations
We introduce the notion of a generalized spin representation of the maximal compact subalgebra $\mathfrak k$ of a symmetrizable Kac--Moody algebra $\mathfrak g$ in order to show that, if defined over a formally real field, every such $\mathfrak k$ has a non-trivial reductive finite-dimensional quotient. The appendix illustrates how to compute the isomorphism types of these quotients for the real $E_n$ series. In passing this provides an elementary way of determining the isomorphism types of the maximal compact subalgebras of the semisimple split real Lie algebras of types $E_6$, $E_7$, $E_8$.