Positive definite hermitian mappings associated with tripotent elements

We give a simple proof of a significant result used by Y. Friedman and B. Russo in 1985, whose proof was originally based on strong holomorphic results. Here we provide a simple proof, directly deduced from the axioms of JB∗∗-triples, of the fact that for each tripotent ee in a JB∗∗-triple EE, the bilinear mapping F1:E1(e)×E1(e)→E2(e),(x,y)↦F1(x,y)={x,y,e}F1:E1(e)×E1(e)→E2(e),(x,y)↦F1(x,y)={x,y,e}, is definite positive (i.e., F1(x,x)≥0F1(x,x)≥0 in the JB∗∗-algebra E2(e)E2(e) and F1(x,x)=0F1(x,x)=0 if and only if x=0x=0), where E1(e)E1(e) and E2(e)E2(e) denote the Peirce-1 and -2 subspaces associated with the tripotent ee, respectively.