Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OP JORDAN TRIPLE SYSTEMS 69
= n Joiβi) = /, βf ...,β,
n n ^
These spaces multiply according to the orthogonality rules
(PI) P(J 0)J 2 = P(J 2)J 0 = {J 0J 2J} = {/ 2e7 0/} - P(Jo)^ - 0 ,
(P2) P(J o)J o c J o, {J^ΛJJ c J lr {J 2 J 2 J o} c J lf
(P3) P(J 2)J 2 + P(J 2)j; + P(J.)Jo + {JM}
whereas we can only say P(J^J λ and P(Ji)J 2 lie somewhere in J.
Proof. Clearly from (2.1), we have a direct decomposition of J
into the sum J/8 7
) of those J (il,...,