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Abstract  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    E  8   , 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.

Existence of Richardson elements for seaweed Lie algebras of finite type