On the definition of quaternionic-linear real matrices

Question

I'm reading Tapp's introduction to matrix groups. The book introduced complex-linear matrices. Let me reproduce the definition in my own words:

Let $B\in M_{2n}(\mathbb R)$. Let $J$ be the matrix $$ J = \left (\begin{array}{} 0 & -I \\ I & 0 \end{array}\right )$$ where $I$ is the identity matrix. Then $B$ is complex linear if and only if $BJ = JB$.

The book goes on to define quaternionic linear real matrices. We now change notation. Let now $I$ denote what we define to be $J$ before and let $J$ now denote the matrix $$ J = \left (\begin{array}{} -I & 0 \\ 0 & I \end{array}\right )$$

Then similar to before we define $B\in M_{4n}(\mathbb R)$ to be quaternionic linear if $BJ = JB$ and $BI = IB$.

My question is: why do we not need to include a case for the quaternionic $k$-component in this definition? (like $KB = BK$?)

Answer 1

$BJ = JB$ and $BI = IB$ implies $BK = KB$. The general fact being used here is that the commutant

$$B' = \{ T \mid BT = TB \}$$

is an algebra, and in particular is closed under multiplication.