Lie algebras are often used to study the algebraic groups from which they originate, but they are interesting objects in their own right as well. Many simple Lie algebras have a particular basis with special properties, invented by Chevalley. It is an extremely useful tool to identify and study Lie algebras.

Algorithms exist, and have been implemented, to compute the Chevalley basis of a Lie algebra that is represented in some way (for example as a structure constant algebra). Unfortunately, these algorithms break down in some special cases and in particular over fields of characteristic 2 and 3.

We show what problems arise in these cases and how they can be solved, and apply the newly found algorithms to the study of Lie algebras generated by extremal elements.