I tried to solve another exercise and I would be grateful if someone could tell me if my answer is right. This is the exercise:

> 
> Characterize the pairs $p,q \in \mathbb H$ such that $pq = -qp$.
> 
> 
> 

I can see that the unit quaternion paris $(i,k), (i,j), (j,k)$ anticommute.

Since the real part of two purely imaginary quaternions is the dot product and the imaginary part the vector product it follows that purely imaginary quaternions anticommute.

If $a+p, b+q$ are two quaternions with real parts $a,b$ then $(a+p)(b+q) = ab + aq + bp + pq$. The only way for this to anticommute is if $ab=0$. Therefore I conclude that all pairs of purely imaginary quaternions anticommute.

Is this correct?