Biderivations of the Parabolic Subalgebras of Simple Lie Algebras

Let 𝔤 be a simple Lie algebra of rank l over an algebraically closed field of characteristic zero, 𝔭 an arbitrary parabolic subalgebra of 𝔤. A bilinear map ϕ: 𝔭 × 𝔭 → 𝔭 is called a biderivation of 𝔭 if it is a derivation with respect to both components, meaning that for all x, y, z ∈ 𝔭. It is shown in this article that a bilinear map ϕ: 𝔭 × 𝔭 → 𝔭 is a biderivation if and only if it is a sum of an inner and an extremal biderivation.