Is there a characterisation of Cayley–Dickson Algebras?

The Cayley–Dickson construction takes an algebra with involution and produces another algebra with involution of twice the dimension.

Starting from the reals (with trivial involution), we progressively get the complex numbers, the quaternions, the octonions and the sedonians. However, we need not stop here, the construction can be iterated ad infinitum.

These algebras need not be associative, however, they are always power-associative. In 1954, Schafer showed that these algebras also satisfy the flexible identity. Moreover, the involution helps defines a norm.

It’s unlikely that these properties characterise these algebras, which for the lack of a name, I’ll call the Cayley–Dickson algebras. Is there a characterisation of these algebras, or some useful results towards this?