I have a few questions concerning Kostant's work on principal three-dimensional subalgebras (TDS). Let $\frak{g}$ be a finite-dimensional complex semisimple Lie algebra, and $\frak{a}\subseteq\frak{g}$ a principal TDS. Is it true that the centralizer $Z_{\frak{g}}(\frak{a})=\{\xi\in\frak{g}:[\xi,\eta]= $0$ \text{ }\forall\eta\in\frak{a}\}$ is trivial (or equivalently, that the trivial one-dimensional representation of $\frak{a}$ is not an irreducible constituent of the $\frak{a}$-module $\frak{g}$)?

Secondly, if we require that $\frak{g}$ be simple with exponents $m_1,\ldots,m_r$ ($r$ is the rank of $\frak{g}$), then the $\frak{a}$-representation $\frak{g}$ decomposes as $$\frak{g}=\bigoplus_{i=1}^{r}\frak{g}_i,$$ where $\frak{g}_i$ is an irreducible representation of dimension $2m_i+1$. To what extent does this also hold for $\frak{g}$ semisimple?

Thanks!