  According to Kirillov's theory, the construction of a unitary irreducible
representation of a nilpotent Lie group requires a precise computation of some
polarizing subalgebra subordinated to a linear functional in the linear dual of
the corresponding Lie algebra. This important step is generally challenging
from a computational viewpoint. In this paper, we provide an algorithmic
approach to the construction of the well-known Vergne polarizing subalgebras
\cite{Corwin}. The algorithms presented in this paper are specifically designed
so that they can be implemented in Computer Algebra Systems. We also show there
are instances where Vergne's construction could be refined for the sake of
efficiency. Finally, we adapt our refined procedure to free nilpotent
finite-dimensional Lie algebras of step-two to obtain simple and precise
descriptions of Vergne polarizing algebras corresponding to all linear
functionals in a Zariski open subset of the linear dual of the corresponding
Lie algebra. Several explicit examples are given throughout the paper. Also, a
program written for Mathematica is presented at the end of the paper.
