Existence of Richardson elements for seaweed Lie algebras of finite type

Seaweed Lie algebras are a natural generalisation of parabolic subalgebras of reductive Lie algebras. A well-known theorem of Richardson says that the adjoint action of a parabolic group has a dense open orbit in the nilpotent radical of its Lie algebra (Richardson, Bull. Lond. Math. Soc. 6 (1974) 21–24.). Elements in the open orbit are called Richardson elements. In (Jensen, Su and Yu, Bull. Lond. Math. Soc. 42 (2009) 1–15.) together with Yu, we generalised Richardson's Theorem and showed that Richardson elements exist for seaweed Lie algebras of type A. Using GAP, we checked that Richardson elements exist for all exceptional simple Lie algebras except E₈, where we found a counterexample. In this paper, we complete the story on Richardson elements for seaweed Lie algebras of finite type, by showing that they exist for any seaweed Lie algebra of type B, C and D. By decomposing a seaweed Lie algebra into a sum of subalgebras and analysing their stabilisers, we obtain a sufficient condition for the existence of Richardson elements. The sufficient condition is then verified using quiver representation theory. More precisely, using the categorical construction of Richardson elements in type A, we prove that the sufficient condition is satisfied for all seaweed Lie algebras of type B, C and D, except in two special cases, where we give a direct proof.