The index of a complex Lie algebra is the minimal codimension of its coadjoint orbits. Let us suppose $\g $ semisimple, then its index, $\ind \g $, is equal to its rank, ${\rm rk}~\g $. The goal of this paper is to establish a simple general formula for the index of $\n (\g ^{e})$, for $e$ nilpotent, where $\n (\g ^{e})$ is the normaliser in $\g $ of the centraliser $\g ^{e}$ of $e$. More precisely, we have to show the following result, conjectured by D. Panyushev Panyushev (2003) : $ \ind \n (\g ^{e}) = {\rm rk}~\g -\dim \z (\g ^{e}), $ where $\z (\g ^{e})$ is the centre of $\g ^{e}$. Panyushev (2003) obtained the inequality $\ind \n (\g ^{e}) \geq \rg \g -\dim \z (\g ^{e})$ and we show that the maximality of the rank of a certain matrix with entries in the symmetric algebra ${\mathcal S}(\g ^{e})$ implies the other inequality. The main part of this paper consists of the proof of the maximality of the rank of this matrix.