Find Cartan Subalgebra for Algebra Given Generators

[CuriosusNN]

Hey! I am studying group theory for particle physicists right now and I came across the following general question (Tell me if you think this rather belongs to the homework section, I am new here.)

Say I am given a set of matrices which represent the generators of an algebra, but I don't know which algebra, i.e. I don't know the commutation relations nor anything like roots or similar about the algebra.
Now I wonder how I could reconstruct the algebra in a smart way. I guess the first step is finding the Cartan subalgebra. Once I have that I should be ready to reconstruct all the rest.
So it all boils down to the question: How can I find the Cartan subalgebra, in particular how can I find out what its dimension (i.e. the rank of the algebra) is, so that I know how many generators I can simultaneously diagonalise?
I suppose this is an undergrad linear algebra question, but I would nevertheless appreciate some hint!

 

[fresh_42]

Algorithms for Lie algebras given the multiplication table are notoriously a bit more complicated than what we have in the rest of linear algebra, especially if the scalar field isn't real or complex. For algebraic closed scalar fields of characteristic not two, the CAS is the maximal toral subalgebra, i.e. the largest subalgebra which can be simultaneously diagonalized in the adjoint representation. It is also Abelian in this case, which is a strong requirement. However, in general we have only nilpotency. Hence an algorithm should look for the semisimple parts of the Jordan-Chevalley decomposition of the adjoint representation.