That is, that $x(yx^2) = (xy)x^2$ ?
In the original paper of A. Albert the proof is based on a direct calculation, so it is not even clear why $dim=3$ is important. References show that since then a rich theory with many generalizations has been developed - unfortunately too rich for a beginner to see the wood for the trees.
Why 3×3 Hermitian matrices over an octonion algebra form a Jordan algebra and what means its exceptionality is explained in a very nice book by Kevin McCrimmon "A taste of Jordan algebras" http://www.amazon.com/books/dp/0387954473 (I think it is possible to find an electronic version on the web). Much shorter review article by the same author "Jordan algebras and their applications": http://www.ams.org/journals/bull/1978-84-04/S0002-9904-1978-14503-0/ is also worth to read.
