It is well known that up to isomorphism, the only three two-dimensional unital associative real algebras are the complex numbers, the split complex numbers, and the dual numbers, corresponding to the polynomial ring quotients $\Bbb R[i]/(i^2+1)$, $\Bbb R[j]/(j^2-1)$, and $\Bbb R[\epsilon]/(\epsilon^2)$, respectively.
Is there a similar characterization for three-dimensional or four-dimensional real algebras? Or for low-dimensional real algebras in general?