Quaternion multiplication question

Question

Take $A \in M_n(\Bbb{H})$ and write it as $A = B + jC$, where $B, C \in M_n(\Bbb{C})$. Show that if $q \in \Bbb{H}$ is of the form $q = u + jv$, with $u, v \in \Bbb{C}$ then $(B + jC)(u + jv) = Bu − C^*v + j(Cu + B^*v)$

I never seen quaternion multiplication before but after googling and finding that j is in $\Bbb{H}$, so according to wikipedia I have the following : $(B + jC)(u + jv) = Bu + Bjv + jCu + j^2Cv = Bu - Cv + j(Cu + Bv)$ - but why does the end term above contain $C^*$ and $B^*$?

Answer 1

Well, you see you have you have $z \in \Bbb{C}$, so $z = a + b*i $

$z*j = a + bij = a + bk\tag{1}$

Now consider $j * z = a - bk$, hence in order to get the result given by $(1)$ we need $z$ to be $z^*$ (its conjugate) and that is how we get our result.

Answer 2

You need to keep in mind that quaternionic multiplication is not commutative. In particular, for $i,j \in \mathbb{H}$, $ij=-ji$. From this you can verify that for $z \in \mathbb{C} \subset \mathbb{H}$, you get $z\cdot j=j\cdot \overline{z}$. In your answer you're assuming, for example, that $jCjv=j^2Cv$. This isn't true due to the non-commutativity, and that is why you're coming up with the wrong answer.