I would like to know if there exists any kind of classification of finite dimensional (possibly non-associative) algebras $A$ over $\mathbb R$ that satisfy the following condition:

There exists a norm on $A$ such that for any $a\in A$ $|a*a|=|a||a|$ (the norm on $A$ is Euclidean).

Of course, complex numbers, quaternions and octonians provide examples of such algebras. But there are further examples of dimension $2^n$ that are given by Cayley-Dickson construction (pages 8-10 in http://math.ucr.edu/home/baez/octonions/octonions.pdf).

Is there a classification of such algebras, at least in small dimensions (over $\mathbb R$)? What are possible dimensions of such algebras?