How to find maximal dimension abelian subalgebra in finite Lie Algebra?

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Is there any well known algorithm how to find maximal dimension abelian subalgebra in finite dimension Lie Algebra? If there is a built-in routine in some computer algebra system, it is the most desirable result...

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Yes, there is. We have worked on such an algorithm; see for example here, and our paper Abelian ideals of maximal dimension for Lie algebras. For semisimple Lie algebras, the maximal dimension of abelian subalgebras is well known and easy to compute. The difficult case is this invariant for solvable Lie algebras, including the construction of such abelian subalgebras.