solve_group receives a list G of infinitesimals corresponding to symmetry generators that generate a finite dimensional Lie Algebra G, and returns a representation of the derived algebras of G.

Derived algebras  of G are defined recursively as follows:

 is G;

 is the Lie Algebra obtained by taking all possible commutators of ;

in general,  is the Lie Algebra obtained by taking all possible commutators of .

Since G is assumed to be finite, there exists a positive integer  with the following properties:

(i)  =

(ii)  is the smallest integer possessing property (i).

solve_group returns a list  of  lists of symmetries with the following properties:

The symmetries inside the list  form the basis for

The symmetries inside the lists  and  together form the basis for .

In general, the symmetries inside the first  lists of  together form the basis for .

In other words, map(op, L[1..n+1-i]) is a basis for .

The group G is solvable if  is the zero group. If G is solvable then the first element of the returned list  will be the empty list [].

This function is part of the DEtools package, and so it can be used in the form solve_group(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[solve_group](..).