Derived filiform Lie algebras having first coefficient \(0\) (Q2762849)

New properties of the invariant \(j\) of complex filiform Lie algebras are found and used in order to get a characterization for a complex filiform Lie algebra \(L\) having the first coefficient \(0\) to be derived. If an adapted base of such an algebra \(\{e_k\mid k=1,\ldots,n\}\), \(n=\dim L\), is chosen and \(j=\max\{k\in\mathbb{N}\mid L^{n-k+1}\text{ is commutative}\}\) is the invariant, \(n=j+h\), \(4\leq h\leq n-j-1\), \(2q<j+7-3h\), then \(L\) is a derived algebra, if \([e_{j+h},e_{j+h+1}]=\lambda_i e_{q+2h}\), with \(1\leq i\leq 3\).