Compatible Peirce decompositions of Jordan triple systems - MSP

76 KEVIN MCCRIMMON
corresponding to (2.7), (2.8).
Proof. & — U if ι remains compatible in any larger J since by
(1.6) compatibility of e,/is an element condition {eef} — P(ff{eef} and
thus remains true in J. Thus S 7 decomposes J with J iό = E iά(J) 3
Eu(J) = Jtj- Since these Peirce projections E ts = Σ-EW •-,**> are
multiplication operators, they leave any ideal K invariant, so K =
Σ •# ^ r
^lse βi _L e3 are orthogonal (so we
have the start ex—e2 of a quadrangle). This latter configuration is
I
very important—it can always be completed to a true quadrangle
where adjacent corners are collinear and opposite corners are orthogonal.
Quadrangles
We define a quadrangle of tripotents to be an ordered quadruple
{e ίf e 2 , e 3 , ej of tripotents such that
(3.1) βi T e i+19 βi J_ e i+2 , {eie i+1 e i+2 } = e i+z (indices mod 4) .
Examples are {E ir , E«, E j8 , E άr ] in M p>q (D) (p, q ^ 2), {F iif F ilf F klt
F kj } in S n (C) (w ^ 4), and {H,,, J5r«, H kl , H kj } in iϊ n (A A, j) (n ^ 4).
QUADRANGLE LEMMA 3.2. e T β T e , e ± e suffices for quad-
x 2 3 λ 3
rangularity: this implies e± = {e^e2ez\ is a tripotent collinear with
eu ez and orthogonal to e2, such that {e2e3e^ = eu {e^e^ — e e2 2y
β3, so {elf e2f e3, e4} is a quadrangle.