I'm trying to understand Quaternions. So I understand that a Quaternion is written like $xi+yj+zk+w$. I also understand that $i^2 = j^2 = k^2 = ijk = -1$, and how that can be used to derive equations such as $ij = k$ and $jk = i$.
One things that confuses me is that $i$ is not equal to $j$ which is not equal to $k$. I can say $i^2 = -1$ and $j^2 = -1$ but I can't say $ij = -1$.
Correct me if I'm misunderstanding something, but why do they seem to have the same product when squared yet all three must be multiplied to equal the product of any one of them squared? Are they supposed to be on different imaginary planes?