Real, complex, and quaternion coefficient algebras and matrix notation

Let $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ be the complex field, whose elements have unique coordinates $z=a+b\mathrm{i}$ (The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$). Write $\bar{z}=a-b\mathrm{i}$ for complex conjugation (Real and imaginary parts, complex conjugation, and modulus). Let

$$\mathbb{H}=\{\left(\begin{array}{cc}z& w\\ -\bar{w}& \bar{z}\end{array}\right):z,w\in \mathbb{C}\}\subseteq M_2(\mathbb{C})$$

with addition and multiplication inherited from matrices. This is a real associative algebra: multiplying the matrices with parameters $(z,w)$ and $(z^{\mathrm{\prime }},w^{\mathrm{\prime }})$ gives the matrix with parameters $(z{z}^{\mathrm{\prime }}-w{\bar{w}}^{\mathrm{\prime }},z{w}^{\mathrm{\prime }}+w{\bar{z}}^{\mathrm{\prime }})$. Its real basis is $1,i,j,k$, where $i=\mathrm{diag}(\mathrm{i},-\mathrm{i})$, $j=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, and $k=ij$; these satisfy $i^2=j^2=k^2=-1$, $ij=k$, $jk=i$, $ki=j$, and reversing any of the last three products changes its sign.

For $D\in \{\mathbb{R},\mathbb{C},\mathbb{H}\}$ and a positive integer $m$, $M_m(D)$ is the associative algebra of $m\times m$ matrices with entries in $D$. The direct sum $A\oplus B$ of algebras has componentwise addition and multiplication. The opposite algebra $D^{\mathrm{o}\mathrm{p}}$ has the same addition and unit as $D$, with multiplication $d{\cdot }_{\mathrm{o}\mathrm{p}}e:=ed$. A left module is simple when it is nonzero and has no proper nonzero submodule (Simple module: a nonzero module with no proper nonzero submodule). For a simple left $M_m(D)$-module $S$, its coefficient type is $D$ when ${\mathrm{End}}_{M_m(D)}(S)\cong D^{\mathrm{o}\mathrm{p}}$; the classification theorem below proves the matrix-algebra cases used on this page.