What are the structure constants for the algebra of quaternions? Show this algebra is associative.
How can I find the structure constants? I know that for an algebra $\mathscr{A}$ and basis $B=\{e_i\}_{i=1}^N$ for the underlying vector space, we have $e_ie_j=\sum\limits_{k=1}^N c_{ij}^k e_k$ for $c_{ij}^k\in\mathbb{C}$, which are the structure constants ($c_{ij}^k$). But I don't know how to find them for quaternions.
Ok, someone explained to me that $c_{ij}^k=c_{11}^1=\{1,0,0,0\}$, for $e_ie_j=e_ie_j$, but I still don't understand how the multiplication works. Can some explain?
Basis: $e_1=\{1,0,0,0\};e_2=\{0,1,0,0\};e_3=\{0,0,1,0\};e_4=\{0,0,0,1\}$.
As requested, I'm posting my comment as an answer. See e.g. here.
The standard basis of the quaternion algebra is $\{1,i,j,k\}$, and we have $i^2 = j^2 = k^2 = -1$, $ij = -ji = k$, $jk = -kj = i$, $ki = -ik = j$.