Indice et decomposition de Cartan d'une algebre de Lie semi-simple reelle

The Iwasawa decomposition $\mathfrak{g}=\mathfrak{k}_0 \oplus \hat{\mathfrak{a}}_0 \oplus \mathfrak{n}_0$ of the real semisimple Lie algebra $\mathfrak{g}_0$ comes from its Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{p}_0$. Then we get $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{b}_0$ where $\mathfrak{b}_0=\hat{\mathfrak{a}}_0 \oplus \mathfrak{n}_0$. The question of knowing if the index were additive in the decomposition $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{b}_0$ goes back M. Ra\"{i}s \cite{Rais}. In \cite{Moreau3}, I wrote that the index always is additive for this decomposition. Precisly, I claim that the index of $\mathfrak{b}$ is given by the following formula : {\rm ind} \mathfrak{b} = {\rm rk \} \mathfrak{g} - {\rm rk \} \mathfrak{k}, where $\mathfrak{b}$ is the complexification of $\mathfrak{b}_0$. This result is false in general. We actually have an inequality : {\rm ind} \mathfrak{b} \geq {\rm rg \} \mathfrak{g} - {\rm rg \} \mathfrak{k}. The goal of this paper is to correct this mistake. We resume the approach of \cite{Moreau3} to obtain this time the previous inequality. Then we give in more a characterization of the semisimple real Lie algebra $\mathfrak{g}_0$ for which the index is additive in the decomposition $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{b}_0$. Moreover, we study in this paper the quasi-reductive character of some subalgebras of $\mathfrak{g}$. This is a new part in comparison with \cite{Moreau3}.