Jordan triple systems and pairs
do not in general possess unit elements, so that certain standard Jordan algebra
methods for studying derivations, extensions, and bimodules do not carry over to
triples. Unit elements usually arise as a maximal sum of orthogonal idempotents. In
Jordan triple systems such orthogonal sums of tripotents are not enough: in order to
“cover” the space one must allow families of tripotents which are orthogonal or
collinear. We show that well behaved triples and pairs do possess covering systems of
mixed tripotents, and that for many purposes such nonorthogonal families
serve as an effective substitute for a unit element. In particular, they can be
used to reduce the cohomology of a direct sum to the cohomology of the
summands.
