Liouville theorem for Jordan algebras

We give a generalization for Jordan algebras of the classical Liouville theorem describing the conformal transformations of a euclidean vector space. In a first step we establish an infinitesimal version which is purely algebraic; namely, we show that the structure Lie algebra of a simple Jordan algebra (not isomorphic to R or C) is of finite order 2. In a second step, using only elementary calculus and Lie theory, we deduce the global version describing the transformations of a simple Jordan algebra which are conformal with respect to the structure group of the Jordan algebra. It turns out that these transformations form a Lie group of birational transformations, also known as the Kantor-Koecher-Tits group, and we can caracterize this group as the group of conformal transformations of the conformal closure of the Jordan algebra.