74 VIII. WIT T INVARIANT S
(2) A differen t proo f o f Propositio n 30.13 was suggeste d b y M . Knus : Writ e
(E ® E)+ fo r th e subalgebr a o f E ® E consistin g o f elements fixe d b y th e
involution whic h permute s th e factor s (i.e. , (E (g ) E)+ i s T S (£") , wher e
TS mean s "symmetri c tensors" , no t t o b e confuse d wit h th e symmetri c
powers, whic h ar e quotients) . Thi s i s a n etal e algebra . Th e obviou s ho -
momorphism E g ) E E restrict s t o a surjectiv e ma p (E ® E)+ » E,
hence E i s a direct summan d o f (E®E)+. Indeed , (E®E)+ i s isomorphic
to E x E(2). Thi s decompositio n i s compatible wit h trac e forms , henc e
TS2(q) = q+(2)-q
E{2)
where TS 2(g) i s the symmetri c squar e o f q, i.e., th e restrictio n o f q (g) q to
the symmetri c par t o f E 0 E. B y takin g orthogona l bases , on e see s tha t
T52(q)=n+(2)-X2(q).
This give s the desire d formul a
te(2)=A2((7)-(2)(g-n).
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