Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 73
I on J x(e) Π JiGO, and J,{e) Π J x(f) c J 2(e + /). (iii) If firle, / then
g±e +/, so then result follows from (i) and (ii). •
Continuing the proof of (2.5), by (2.9i) an element in 2(
automatically lies in J o( i?i) for all j Φ i by orthogonality of g 7 *,
so J i4 = J 2( g 7 ,) c Πί>ί Jo( &j) is the sum
J a — ZJL J ιo,- ,o\i v
,2 t' ',i m\Q, ",o)
of those Peirce spaces in (2.1) having at least one 2 in the ΐth
range of indices (hence O's in all other ranges).
By (2.9iii), an element cannot belong to three different J ι(& τ),
so ^(g?) - ΣiΛ(8y Π ΓWΛ(ίfi) + Jo(&i)} is the sum
of those Peirce spaces in (2.1) having no 2's and at least one 1 in
the ith range but no l's (only O's) in the other ranges, together
with the sum
of those Peirce spaces in (2.1) having no 2's but at least one 1 in
the ith and jfth ranges (hence O's in all other ranges).
Finally, Jo(&) = Π Jo(i?«) is
the Peirce space with no 2 ?
s or l's,
only O's:
^oo = = J(o, ,o; ;o, ,o)
This yields the decomposition J — 0 J and the expression for
i3
Jfc(if) and Jk {^i) in terms of the Jiά .
Most of the Peirce relations follow directly from (2.2) (Pl-3) in
the form of the rules
(A) P{Jio )Jkl c Σ {JPq IP, ? 6 {i, i, fc, ϊ, 0}}
{ΛiΛi Jmn) c Σ {«/" OT IP, ? 6 {i, i, fc, ϊ, m, n, 0}}
(B) P(Jr8 )J or {JrsJJ} has no component in Jtu for
{*, w} Π {r, s} = 0
(C) {Joi&rWoi&rWii&r)}