What are the structure constants for the algebra of quaternions? Show this algebra is associative. 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
$i^2 = j^2 = k^2 = -1$, $ij = -ji = k$, $jk = -kj = i$, $ki = -ik = j$. 
