With u = 1,2,3,4 and ',' denoting
partial differentiation so ,u means \frac{\partial }{\partial x^{u}}
The commutator structure of this group can be written as:
\zeta^{u}(x) = \epsilon^{i}f^{u}_{i}-----i = 1,2,...,N,-----(1)
Where the \epsilon^{i} are the group parameters. Our basic requirement is that the commutator of two such infinitesimal mappings must again be of this form. We find for this commutator:
\zeta^{u}_{3} = (\epsilon^{i}_{2}\epsilon^{j}_{1}-\epsilon^{i}_{1}\epsilon^{j}_{2})f^{u}_{i,v}f^{v}_{j}-----(2)
In order that it be of the form (1) the functions f^{u}_{i} must be related to each other by an equation of the form:
f^{u}_{i,v}f^{v}_{j} = c^{k}_{ij}f^{u}_{k}-----(3)
Where the c^{k}_{ij}are constants independent of the \epsilon^{i} and the x^{u} .
they are called the structure constants of the group and serve to characterize it in
a manner that is independent of the particular form taken by the f^{u}_{i}.
If we substitute (3) back into (2) we obtain:
\zeta^{u}_{3} = (\epsilon^{i}_{2}\epsilon^{j}_{1}-\epsilon^{i}_{1}\epsilon^{j}_{2})c^{k}_{ij}f^{u}_{k}
So that the infinitesimal parameters \epsilon^{k}_{3} of the commutator are given by:
\epsilon^{k}_{3} = c^{k}_{ij}(\epsilon^{i}_{2}\epsilon^{j}_{1}-\epsilon^{i}_{1}\epsilon^{j}_{2})
From the manner of their construction we see that the structure constants
are antisymmetric in the two lower indices, that is,
c^{k}_{ij} = -c^{k}_{ji}