Structure constant ($c_{ij}^k$)

Choose a basis $(e_i{)}_{i=1}^n$ of a finite-dimensional algebra over a field $k$. Its structure constants are the coefficients $c_{ij}^r\in k$ determined byMultiplication is a bilinear map, so these finitely many coefficients determine every product: if $x={\sum }_i{x}_i{e}_i$ and $y={\sum }_j{y}_j{e}_j$, then the $r$th coordinate of $xy$ is ${\sum }_{i,j}{c}_{ij}^r{x}_i{y}_j$. They turn abstract multiplication into explicit polynomial coordinate functions.

In the complex numbers viewed as a real algebra over a field, use the basis $e_1=1,e_2=i$. The nonzero structure constants are $c_{11}^1=c_{12}^2=c_{21}^2=1$ and $c_{22}^1=-1$. Changing the basis changes these numbers without changing the algebra isomorphism class. Associativity becomes a checkable family of polynomial equations:Both sides are the $e_s$ coefficient of the two ways of associating $e_i{e}_j{e}_k$.