Four dimensional absolute valued algebras containing a nonzero idempotent commuting with all idempotents

In [9], we have proven that if A is a four-dimensional absolute valued algebra having two different subalgebras isomorphic to C , then A is isomorphic to H , M 1 , M 2 or M 3 . Here we complete the study of A . Indeed, we show if A has two different subalgebras isomorphic to ∗ C . Then A is isomorphic to ∗ H , ∗ M 1 , ∗ M 2 or ∗ M 3 . Furthermore, we classify all four-dimensional absolute valued algebras containing a nonzero idempotent commuting with all idempotents, such an algebra A contains at least two different subalgebras isomorphic to ∗ C . Which means that A is isomorphic to ∗ H , ∗ M 1 , ∗ M 2 or ∗ M 3 .