Suppose $\mathcal{A}$ is a unital associative algebra over $\mathbb{R}$. If we identify $\mathcal{A} = \mathbb{R}^n$ then the $\mathcal{A}$ multiplication corresponds to particular linear maps on $\mathbb{R}^n$. Of course any linear map on $\mathbb{R}^n$ corresponds uniquely to its standard matrix hence we obtain a correspondence between vectors in $\mathcal{A}$ and matrices in $\mathbb{R}^{n \times n}$. These square matrices are known as the **left regular representation** of the algebra. This is not unique unless we add additional data about the correspondence of the $n$-dimensional algebra and its presentation on $\mathbb{R}^n$.

My favorite examples, $\mathbb{C} = \mathbb{R}^2$ is naturally identified with the subalgebra of $2 \times 2$ matrices of the form:

$$ \left[\begin{array}{cc} a &-b \\ b &a \end{array} \right] $$

Or the hyperbolic numbers $\mathbb{R}+j\mathbb{R}$ identified with

$$ \left[ {\begin{array}{cc} a & b \\ b & a \end{array}} \right] $$

Or the direct product of $\mathbb{R}$ with $\mathbb{R}$ the element $(a,b)$ is identified with

$$ \left[ \begin{array}{cc} a & 0 \\ 0 & b \end{array} \right] $$

Up to isomorphism the last two examples are actually the same example. I know of one other two-dimensional algebra up to isomorphism.

Question:where can I find a complete tabulation of the low-dimensional left-regular representations of unital algebras?

I have found many results on google and here, but I can't find one which stands out as a continuation of the list I began at the start of this post. In particular, complete list of complex associative algebra, is great except the base-field is $\mathbb{C}$. If there was a simple theorem that allowed me to extract the list I desire from that list then that would also be a useful answer. But, I'd rather have direct reference for a list of the real associative left regular representations. Ideally this will help me choose a good notation if there already is an agreed notation accepted among those who worked on such classifications.

As always the help of the MO community is greatly appreciated.