Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 77
Proof. One can verify this directly, or use the exchange
automorphisms of [7] taking e u e 2 , e z , e±-*e 2 , e u —e i9 — e z and e u e 2 , e z ,
e,--> -e i9 e 3 , e 2 , -e γ . Π
The Peirce decompositions (2.1) and (2.2) simplify in the case
of a quadrangle
3.3. Quadrangular Decomposition. // g 7
= {elf e2 , ez , ej is a
quadrangle of trίpotents in J then the multiplication operators
satisfy
( i ) L(eit ei+ ι) = L(ew , eί+2 ) (indices modulo 4)
(ii) Pfe)Pfe +1) = P(ew)P(ew) (iii) L(e l9 e x ) - L(e 2 , e 2 ) + L(β 8 , β 3 ) - I^(β 4 , β 4 ) = 0.
Peirce decomposition relative to & is J =
==
^ ) =:
{^(2200) + ^(0022) + «^(2002) + ^(0220)}
~ ^(1210) + e/(0121)J
while all other Peirce spaces vanish.
Proof. For (i) we have Lfe, e