Compatible Peirce decompositions of Jordan triple systems - MSP

COMPATIBLE PEIRCE DECOMPOSITIONS OF JORDAN TRIPLE SYSTEMS 85
in (iv') are L(J ιm)t J ιm))(J mi) + Jam) c J m) + J (m) where J (200) = 0
again by (3a), L(J {211), J i220))Jam c J( m), L(J (220), J( 2n))Jaoi) c/mo). Thus
the conditions of 4.3 are met, and S& is an automorphism in this
case.
Now consider the case g 7 = {e, /}. We first note that conditions
(2a)-(2c) imply the further conditions
(2a'-b') P(J {ll)){J m + Jim + Jao) + J m) + e/coo)} = 0
(2C') P(/ (]0), e/ (01)){e7 (11) + J (oo)} = 0
(2C ) Z/(e/ (11), e/ (10) + J(0l))J(00) ~ 0
(2C ) L(J (10)f J(n))J(12) — L(Jm)t J(ii))e/ (21) = 0.
Indeed, (2a-b) imply (2a'-b') since P(a?u)»ii={{a?πtfii/}/a?ii}-{(P(ί»u)/)»ii/}
(by (0.4)) = 0 by (2b) and {x11yjιf} e J2_j)2 = 0 for j = 2 or 0 by (2a).
(2c) implies (2c') since {x10y00z01} — {tfioί^ooW*}/} (by linearized (0.3)
acting on x10) = {x10wuf} = 0 by (2c), and {α? loyu«oi} = {&ioβ{eyiA>i}} (by
linearized (0.3) on a? ι0) = 0 by (2c), (2c) implies (2c") since {zQOy1Qxn} =
{^oo{2/io^ii/}/} (by linearized (0.3)) = 0 by (2c), and (2c) implies (2c'")
since {x^Vn^n} = {{VioVuf}f*it} (by linearized (0.3)) = 0 by (2c) again.
Thus we may employ all these.
To verify (4.3(i)-(iv)) for g 7 = {e, /} it suffices by symmetry in
e,/and UJI9JX)JK(ZJK to verify (i'-ii') P(/ (11)), P(Jm), P(J {W, Jm), P(J(io), Jm)* P(J(2»)> P(J{u), Jm) a H leave Jt and J2 + Jo invariant,
(iii') L(Jm), J (10)) and L(Jm, J (w) and L(J (10), J (01)) leave J2 + Jo invariant,
(iv') L(J (21), J (12)) leaves Jx invariant. By the Peirce relations
(1.2) the only nontrivial products in (i'-ii') are P(Jai))({J{2u + J^m +
Jm)) + {Jtu) + Jm + Jm)}) c {0 + 0 + 0} + {Jai) + 0 + 0} by (2a' - b'),
P(Jao^Jm) + ^do)) c 0 + J (10) by (2a), P(Jai), Jm)({Jm) + Jm] +
{Jtn) + Jm + Jm)}) C {Jm + J (21)} + {J (10) + Jai) + 0} by (2a), P(Jm, J Γ
(oi))(e/(oo) + {J Γ
(π) + J r
(lo)+J (o1)})cO + {O + J (ol) + JΓ (10)} by (2c'), P(e7 (21))J
(21)c
J (21), P(J (21), J r
(i2))({/ (2i) + J (12)} + {J (11)}) c {/(!,) + J (21)} + 0 by (2a); in
(iii') are L(Jin), Jm)(J