Four dimensional absolute valued algebras having two different subalgebras isomorphic to C

The aim of this paper is to construct, by algebraic methods, some new class of four-dimensional absolute valued algebras, namely M 1 , M 2 and M 3 which are not isomorphic to the known algebra H . These new algebras contain a nonzero omnipresent idempotent. Furthermore, we classify all four-dimensional absolute valued algebras having two dif-ferent subalgebras isomorphic to C . Note that there exists a four-dimensional absolute valued algebra containing no subalgebra of dimension two, which means that, the problem of classifying all four-dimensional absolute valued algebras seems still to be open.