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 Euclidean norm on $A$ such that for any $a\in A$ $|a\*a|=|a||a|$.


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? 



A classification of non-associative algebras with a norm?
I don't think these objects can be classified in a manner similar to the normed unital division algebras, if you take "algebra" to mean "vector space $V$ equipped with a bilinear map $V \otimes V \to V$". In particular, I suspect you end up with high-dimensional moduli spaces of such structures in all large dimensions.


Here is a naive calculation of degrees of freedom:


Let $a\_{i,j}^k$ be the structure constants of our algebra, assembled into matrices $A^k$, and consider a point $x = (x\_1, \ldots, x\_n)$. We may write $x \ast x = (x^T A^1 x,\ldots, x^T A^n x)$. The length condition becomes $\left(\sum\_i x\_i^2\right)^2 = \sum\_k \left(x^T A^k x \right)^2$. Writing this out in terms of the coordinates of $x$, we obtain an identity of homogeneous polynomials in $x\_i$ of total degree 4, with coefficients that are quadratic in the structure constants. In other words, the space of solutions is an intersection of $\binom{n+3}{4}$ quadric hypersurfaces in $n^3$-dimensional space.


[Revised following YCor's comment:] When we account for the $O(n)$ symmetry of the solution space, we get the formula
$$ n^3 - \binom{n+3}{4} - \binom{n}{2}$$
which is positive for $2 \leq n \leq 16$ with maximum 299 at $n=13$. When $n$ is large we get more constraints than variables.


Since the Cayley-Dickson construction exists and provides solutions in arbitrarily large dimension, it is clear that the constraints are highly non-generic. This does not completely eliminate the possibility that in some dimensions there are no solutions, but I think it is at least discouraging as far as classification is concerned.


