[knappa]

Unless you mean something different from the usual multilinear algebra meaning of "exterior algebra", xy^yz is zero. (It contains two y's.) I think that you must actually mean Clifford algebras. (But there are also signs when in characteristic ≠ 2.)

[blurbleblurble]

Yeah, I do mean Clifford algebras. Thanks for the catch. I'll have to study up on the distinction between "outer products" in Clifford algebras and exterior algebras.

[johncolanduoni]

They are isomorphic as vector spaces (again with characteristic != 2), but their products are not preserved by the isomorphism (unless the Clifford algebra is trivial).