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

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

Thanks!