A classification of nilpotent compatible Lie algebras

Key Points

• A full classification reveals 3 isomorphism classes and a one-parameter family for nilpotent compatible Lie algebras in dimension 3.
• In dimension 4, there are 10 isomorphism classes, 7 one-parameter families, and 2 two-parameter families identified.
• Using the extended Skjelbred-Sund method, compatible Lie algebras are classified over fields of characteristic different from 2, showcasing distinct structural properties of nilpotent types. The findings of this classification offer new insights into the organization of compatible Lie algebras, though further exploration in higher dimensions may be needed.