Lie algebra Lie algebra ideal A Lie algebra ideal π β΄π€ is just an algebra ideal of a Lie algebra. lie Equivalently, π β΄π€ is a submodule under the adjoint representation. Note that a Lie subalgebra is a (two-sided) ideal iff it is a left or right ideal, by the alternating property. A Lie algebra is simple iff it has no nontrivial ideals. Properties Let π,π β΄π€ be ideals. Then π +π is an ideal π β©π is an ideal [π,π] is an ideal (see Commutator ideal) Proof ^P1 and ^P2 follow immediately. Note that for any π₯ βπ€, by ^P1 adπ₯β‘[π,π]=[adπ₯β‘π,π]+[π,adπ₯β‘π]=[π,π] so [π,π] is an ideal. Special ideals Centre of a Lie algebra Kernel of a Lie algebra homomorphism Central ideal Radical of a Lie algebra See also Quotient Lie algebra tidy | en | SemBr