Upper bound for the dimension of the center of a solvable Lie algebra (Q1866801)

According to the Levi decomposition theorem, every solvable Lie algebra \(L\) over a field \(K\) of characteristic zero is a semi-direct sum of its radical \(\text{Rad}(L)=R\) and \(S\cong L/R\), where \(S\) is semisimple. \(S\) has no center and consequently \(Z(L)\subset R\). Thus the study of \(Z(L)\) is reduced to the case \(S=0\), i.e. to the case of solvable Lie algebras. It is shown that the dimension of \(Z(L)\) of finite \(n\)-dimensional solvable Lie algebra having a nilradical of dimension \(r\) admits \(2r-n\) as least upper bound.