TITLE: Information about an algebraic group: is it finite?
QUESTION [0 upvotes]: Recently I looked an example of an algebraic group as follows. Consider $\mathbb{H}_{\mathbb{R}}$, the Hamiltonian quaternion algebra over $\mathbb{R}$. Then $\{1,i,j,k\}$ is a basis of this vector space over $\mathbb{R}$. 
Let $R_i, R_j,R_k$ denote the operators on $\mathbb{H}_{\mathbb{R}}$ which are multiplication by $i$, $j$ and $k$ respectively. Then they will give three matrices over $\mathbb{R}$ of size $4\times 4$. Consider the set 
$$G=\{g\in {\rm GL}_4(\mathbb{R}) : gR_i=R_ig, gR_j=R_jg, gR_k=R_kg\}.$$
Then $G$ is an algebraic group. 
I was considering whether this is a finite group, and my intention was that it corresponds to centralizer of $\{i,j,k\}$ in $\mathbb{H}_{\mathbb{R}}$, or equivalently center of $\mathbb{H}_{\mathbb{R}}$, which is $\mathbb{R}$ only, so $G$ is trivial group. Am I right? If not, my next question is - is the algebraic group $G$ finite?

REPLY [0 votes]: It is clearly infinite, the matrix $rI_4$ belongs to it for any non-zero real $r$.
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