On the length of the spectral sequence of a Lie algebra extension. II (Q1841830)

The author [\textit{D. W. Barnes}, Proc. Am. Math. Soc. 129, No. 2, 347-350 (2001; Zbl 0972.18009)] had found sharper bounds for the length \(l(V)\) of the Hochschild-Serre spectral sequence of a Lie algebra extension acting on a module \(V\), with some restriction on the Lie algebra extension and on the module \(V\). In the same paper some questions are also asked. Namely: What happens if we weaken the conditions for sharper bounds? If the module of coefficients \(V\) has the submodule \(W\), is the length \(l(V)\) bounded by some function of \(l(W)\) and \(l(V/W)\)? In the present paper the author gives us an example which shows that the conditions cannot be weakened and \(l(V)\) is not bounded by any function of the lengths \(l(W)\) and \(l(V/W)\).