  We investigate the real Lie algebra of first-order differential operators
with polynomial coefficients, which is subject to the following requirements.
(1) The Lie algebra should admit a basis of differential operators with
homogeneous polynomial coefficients of degree up to and including three. (2)
The generator of the algebra must include the translation operators
$\partial_k$ for all the variables $x_1$,...,$x_k$. (3) The Lie algebra is the
smallest indecomposable Lie algebra satisfying (1) and (2). It turns out to be
a 39-dimensional Lie algebra in six variables ($k=6$) and the construction of
this algebra is also the simplest possible case in the general construction of
the Lie algebras of the transitively differential groups introduced by
Guillemin and Sternberg in 1964 involving the coefficients of degree 3. Those
algebras and various subalgebras have similarities with algebras related to
different applications in physics such as those of the Schr\"odinger, Conformal
and Galilei transformation groups with and without central extension. The paper
is devoted to the presentation of the structure and different decompositions of
the Lie algebra under investigation. It is also devoted to the presentation of
relevant Lie subalgebras and the construction of their Casimir invariants using
different methods. We will rely, in particular, on differential operator
realizations, symbolic computation packages, the Berezin bracket and virtual
copies of the Lie algebras.
