Classification of filiform Lie algebras in dimension 8 (Q1092993)

We give the classification of complex nilpotent filiform Lie algebras in dimension 8. Recall that an \(n\)-dimensional nilpotent Lie algebra \(\mathfrak g\) is filiform if there is a vector \(X\) in \(\mathfrak g\) and a basis \((X,X_ 2,\ldots,X_ n)\) such that \((\text{ad}\, X)(X_ i)=X_{i-1}\), \(i=3,\ldots,n\). We prove that we have a filiform 8-dimensional Lie algebra whose orbit is open in the manifold of the 8-dimensional nilpotent Lie algebra structure.