A generalization of the Kantor-Koecher-Tits construction

The Kantor-Koecher-Tits construction associates a Lie algebra to any Jordan algebra. We generalize this construction to include also extensions of the associated Lie algebra. In particular, the conformal realization of so(p+1,q+1) generalizes to so(p+n,q+n), for arbitrary n, with a linearly realized subalgebra so(p,q). We also show that the construction applied to 3x3 matrices over the division algebras R, C, H, O gives rise to the exceptional Lie algebras f4, e6, e7, e8, as well as to their affine, hyperbolic and further extensions.