TITLE: reference for list of left-regular representations of real associative algebras
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: (Too long for a comment). More (modern and not-so-modern) references, some of them may (partially) contain list(s) you are interested in:

A.A. Albert, Structure of Algebras, AMS, 1939: on page 172 discusses classification of 4-dimensional algebras.
S.C. Althoen, K.D. Hansen, L.D. Kugler, 
A survey of four-dimensional C-associative algebras,
Algebras Groups Geom. 21 (2004), N1, 9-27
R. Ballieu, 
Anneaux finis; systèmes hypercomplexes de rang trois sur un corps commutatif,
Ann. Soc. Sci. Bruxelles Sér. I. 61 (1947), 222-227: presumably contains classification of complex 3-dimensional algebras.
W.A. de Graaf, Classification of nilpotent associative algebras of small dimension,
arXiv:1009.5339.
D. Happel, 
Klassifikationstheorie endlich-dimensionaler Algebren in der Zeit von 1880 bis 1920, 
Enseign. Math. 26 (1980), 91-102 DOI:10.5169/seals-51060 :
A nice historical survey from the modern viewpoint, with a large bibliography.
O.C. Hazlett, 
On the classification and invariantive characterization of nilpotent algebras
Amer. J. Math. 38 (1916), N2, 109-138 http://www.jstor.org/stable/2370262
G. Pickert, Dreidimensionale assoziative nichtkommutative Algebren,
J. Algebra 234 (2000), N2, 280-290 DOI:10.1006/jabr.2000.8550
Scorza, Atti Acad. Sci. Fis. Mat. Napoli 20 (1935), N13 and N14: Classification of 3- and 4-dimensional algebras.
D.A. Suprunenko and R.I. Tyshkevich, Commutative Matrices, 
Acad. Press, 1968 (translation from Russian): On p.61 (of the Russian edition)
there is a discussion of commutative nilpotent algebras of dimension 5.