Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 99
into M {tv ... ίin) .
Proof. For a single tripotent n — 1 this is just (7.7.) Assume
the result for n — 1 tripotents, so we can modify S by something
inner and assume from the start that S(βt), S*(^) lie in M 2 {e*) for
i = 1, 2, , n - 1. We must modify S to obtain S'(e u ), S'*(e n ) e
M z (e n ) without unduly disturbing the previous action on e lt •••, e n _ x .
So consider S' - S - S o for S o = L(m lf e) + L{e, m?) as in (7.7) for
S(e) = m 2 + mi, S*(e) = m 2 * + m? where we set e = β Λ . We know
iS 0 is inner (resp. standard inner derivation) and S'(e), S'*(e) eM 2 (e).
We must show that we haven't disturbed the previous actions,
( * ) Sotei) = {m^eβt} + {emUi} 6 AΓ 2 (e 4 ) (i = 1, -, n - 1)
and dually for So *.
By the induction hypothesis, S preserves Peirce spaces relative
to et . By compatibility, the same is true of L(e, e) and P(e) 2
. Now
(7.1) yields the general identities
[S, L(x, y)] - L(Sx, y) - L(x, S*y)
[S, P(x)P(y)] - P(Sx, x)P(y) - P(x)P(S*y, y)
for structural S, so for x ~ y = e we see
L(m, β) - L(e y m*) = L(mi, β) - L(^, mf)
P(m, e)P(e) - P(e)P{m*, e) = P(m x , e)P(e) - P{e)P(mΐ y e)
preserve Peirce spaces (using m? = m 2 by (7.7i) and (1.4) to get rid
of m 2 ). But then P(m l9 e)P(e)-P{e)P{mΐ t e)-{L{m lf e)-L(e, mΐ)}L(e, e)
also preserves Peirce spaces, and by (0.6) this equals {L(e, e)L(m lf -e) —
L(e, P{e)m 1 )} - {L(e, mΐ)L{e, e) - L(P(e)m*, e)} - L(m u e)L(e, e) + L(e,
mΐ)L(e, e) = L(e, e)L(m l9 e) — L(m lf e)L(e, e) = L({eem^, e) — L{m 1 , {eee})
(by (0.5)) = —L(m u e). Once this preserves Peirce spaces, so does
L(e, mΐ) by (**):
^***^ L(m ίy e), L(e, m*) preserve Peirce spaces J fc fe) .
Thus βieJiiβi) implies {m^e,}, {emf β