I was looking in the wikipedia article about Non-associative Algebra and came across this interesting line:
Each of the properties associative, commutative, anticommutative, Jordan identity, and Jacobi identity individually imply flexible.
It was quite easy to find why the first three properties imply flexible algebra. I was wondering about the last two (Jacobi and Jordan). I went to look for the proof in Schafer but either I missed it or it was given in terms I could not understand.
could someone give a proof similar to the commutative proof?
Thanks!
EDIT: I'm looking for proofs of the following statements:
1) given Jacobi identity prove flexible:
$ \forall x,y,z: (xy)z+(yz)x+(zx)y=0 \Rightarrow \forall x,y: x(yx)=(xy)x $
2) given Jordan identity prove flexible:
$ \forall x,y: (xy)x^2=x(yx^2) \Rightarrow \forall x,y: x(yx)=(xy)x $
EDIT 2: OK, so it appears that Schafer does have a proof for the jordan identity implying flexiblity on chapter V. once I can understand it I hope I can write it here.