  We provide the analytic expressions of the totally symmetric and
anti-symmetric structure constants in the $\mathfrak{su}(N)$ Lie algebra. The
derivation is based on a relation linking the index of a generator to the
indexes of its non-null elements. The closed formulas obtained to compute the
values of the structure constants are simple expressions involving those
indexes and can be analytically evaluated without any need of the expression of
the generators. We hope that these expressions can be widely used for
analytical and computational interest in Physics.
