Let $D \subset \mathbb{C}^n$ be a bounded symmetric domain. It is known that $D$ can be realized as the unit ball of some complex norm $||\cdot||$. Using the Bergman metric on $D$, one can define a triple product on $\mathbb{C}^n$ which enjoys many nice properties. In the paper <http://www.springerlink.com/index/FN661331558V3870.pdf>, a precise formula for an automorphism of such a domain is stated in terms of this triple product (equation 1.8). The reference given in the paper deals with bounded symmetric domains in Banach spaces and involves a lot of machinery. Does anyone know any reference where the automorphisms are described in the finite-dimensional case with proof? I have already had a look at the standard reference <http://molle.fernuni-hagen.de/~loos/jordan/archive/irvine/index.html>, but it does not seem to have it.



Automorphisms of bounded symmetric domains
In S. Helgason's oldest large book, "Differential geometry and symmetric spaces" (?) from the 1960s, symmetric spaces and bounded symmetric domains are discussed in detail. Also, I. Satake's book from c. 1980, I think in a style of the sort you're wanting.


