With the same hypotheses as in Exercise 7, let x⊤y=xy+yx; then the law ⊤ is commutative and doubly distributive with respect to addition but not in general associative.
(a) For all x∈E, show that ⊤m+nx=(⊤mx)⊤(⊤nx).
(b) If we write [x,y,z]=(x⊤y)⊤z−x⊤(y⊤z) (deviation from associativity of the law ⊤), prove the identities
[x,y,z]+[z,y,x]=0
[x,y,z]+[y,z,x]+[z,x,y]=0 [x⊤y,u,z]=[u,[(x⊤y),z)]] (the notation
[x,y] meaning the same as in Exercise 7)
[x⊤y,u,z]+[y⊤z,u,x]+[z⊤x,u,y]=0.