Let $Cl_n$ denotes the clifford algebra generated by quadratic form of signature $(n,0)$. Since $Cl_n$ is either of the form $K(2^m)\oplus K(2^m)$ or $K(2^m)$ where $K=R,C,H$, real, complex and quaternion respectively and $K(2^m)$ is ring of $2^m\times 2^m$ matrices with entries in $K$.
"Let $K_n=R,C$ or $H$ denote the maximal commuting subalgebra for an irreducible real representation of $Cl_n$"
$\textbf{Q:}$ What is the meaning of the sentence here? Is $K_n$ identified in the image of representation or just in $Cl_n$? Let $W$ be a left $Cl_n$ module. Then $Cl_n\to End_R(W)$ defines representation of $Cl_n$ where $End_R(W)$ is $R-$linear endomorphism ring. Since $Cl_n$ is either $K(2^m)$ or $K(2^m)\oplus K(2^m)$ with $K=R,C,H$, it is clear that $R,C$ could be maximal commuting subalgebra for $Cl_n$. I do not see why $H$ could be maximal commuting subalgebra as $H$ is non-commutative.
Ref. Spin Geometry, Lawson Chpt 1, Sec 5.