Finite-dimensional transposed Poisson algebras are nilpotent exactly when left multiplications are nilpotent in both associative and Lie operations, with the nilpotent radical equaling the associative radical and Frattini subalgebra contained in the derived algebra.
Towers, Nilpotency, solvability and Frattini theory for Poisson algebras,J
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Nilpotency and Frattini theory for transposed Poisson algebras
Finite-dimensional transposed Poisson algebras are nilpotent exactly when left multiplications are nilpotent in both associative and Lie operations, with the nilpotent radical equaling the associative radical and Frattini subalgebra contained in the derived algebra.