  We consider the Schatten spaces S^p in the framework of operator space theory
and for any $1\leq p\not=2<\infty$, we characterize the completely
1-complemented subspaces of S^p. They turn out to be the direct sums of spaces
of the form S^p(H,K), where H,K are Hilbert spaces. This result is related to
some previous work of Arazy-Friedman giving a description of all 1-complemented
subspaces of S^p in terms of the Cartan factors of types 1-4. We use operator
space structures on these Cartan factors regarded as subspaces of appropriate
noncommutative L^p-spaces. Also we show that for any $n\geq 2$, there is a
triple isomorphism on some Cartan factor of type 4 and of dimension 2n which is
not completely isometric, and we investigate L^p-versions of such isomorphisms.
