(n– 4)-filiform lie algebras

Breadth and characteristic sequence of nilpotent Lie algebras

“…with δ 2 R,µ 0 (ϕ) = µ 2 0 • 1 ϕ+µ 0 • 1 ϕ• 1 µ 0 +ϕ• 1 µ 2 0 = 0. But any linear deformation of µ 0 associated with a such cocycle belong to F n,3 that is the family of 3-step nilpotent Lie algebras and not necessarily in the sub-family F n, 3 1,1,n−5 of 3-step nilpotent Lie algebras with characteristic sequence (3, 2, 1, · · · , 1).…”

“…From the construction of the linear deformation between any Lie algebra of F n,3 1,1,n−5 and µ 0 , we can assume that ϕ(X 1 , X) = 0 for any X. Now, since µ = µ 0 + ϕ ∈ F n, 3 1,1,n−5 , we have necessarily ϕ(X i , X j ) ∈ K{X 3 , X 4 , X 6 }. The previous identities and lemma imply that ϕ satisfies:…”