Let $\mathfrak{g}$ be a Lie subalgebra of the real upper triangular matrices (i.e. $\mathfrak{g}$ is a completely solvable real Lie algebra). $\mathfrak{g}$ has an exponential radical $\mathfrak{r}$, which is the intersection of the terms of its lower central series. Is it true that any element of $\mathfrak{g}\setminus \mathfrak{r}$ is contained in some Cartan subalgebra? Or more generally is there a necessary and sufficient condition on $\mathfrak{g}$ so that this holds?

The closest result we found is in Bourbaki (Groupes et algebres de Lie, chapitre 7, §3, Proposition 4), stating: let $\mathfrak{g}$ be a Lie algebra of rank $l$ over a field of characteristic $0$. Let $c$ be the nilpotency class of its Cartan subalgebras, and $X\in \mathfrak{g}$. There exists a subalgebra of dimension $l$ with nilpotency class at most $c$ containing $X$.

It is worth mentioning a result about regular elements. An element is called regular if its centralizer has dimension equal to the rank of $\mathfrak{g}$. In that case, this centralizer is a Cartan subalgebra, and every Cartan subalgebra arises like this.

Let $\mathfrak{g}$ be a Lie subalgebra of the real upper triangular matrices (i.e. $\mathfrak{g}$ is a completely solvable real Lie algebra). $\mathfrak{g}$ has an exponential radical $\mathfrak{r}$, which is the intersection of the terms of its lower central series. Is it true that any element of $\mathfrak{g}\setminus \mathfrak{r}$ is contained in some Cartan subalgebra? Or more generally is there a necessary and sufficient condition on $\mathfrak{g}$ so that this holds?

The closest result we found is in Bourbaki (Groupes et algebres de Lie, chapitre 7, §3, Proposition 4), stating: let $\mathfrak{g}$ be a Lie algebra of rank $l$ over a field of characteristic $0$. Let $c$ be the nilpotency class of its Cartan subalgebras, and $X\in \mathfrak{g}$. There exists a subalgebra of dimension $l$ with nilpotency class at most $c$ containing $X$.

Let $\mathfrak{g}$ be a Lie subalgebra of the real upper triangular matrices (i.e.\ $\mathfrak{g}$ is a completely solvable real Lie algebra). $\mathfrak{g}$ has an exponential radical $\mathfrak{r}$, which is the intersection of the terms of its lower central series. Is it true that any element of $\mathfrak{g}\setminus \mathfrak{r}$ is contained in some Cartan subalgebra? Or more generally is there a necessary and sufficient condition on $\mathfrak{g}$ so that this holds?

The closest result we found is in Bourbaki (Groupes et algebres de Lie, chapitre 7, $\S$3, Proposition 4), stating: let $\mathfrak{g}$ be a Lie algebra of rank $l$ over a field of characteristic $0$. Let $c$ be the nilpotency class of its Cartan subalgebras, and $X\in \mathfrak{g}$. There exists a subalgebra of dimension $l$ with nilpotency class at most $c$ containing $X$.

It is worth mentioning a result about regular elements. An element is called $\textit{regular}$ if its centralizer has dimension equal to the rank of $\mathfrak{g}$. In that case, this centralizer is a Cartan subalgebra, and every Cartan subalgebra arises like this.

Let $\mathfrak{g}$ be a Lie subalgebra of the real upper triangular matrices (i.e. $\mathfrak{g}$ is a completely solvable real Lie algebra). $\mathfrak{g}$ has an exponential radical $\mathfrak{r}$, which is the intersection of the terms of its lower central series. Is it true that any element of $\mathfrak{g}\setminus \mathfrak{r}$ is contained in some Cartan subalgebra? Or more generally is there a necessary and sufficient condition on $\mathfrak{g}$ so that this holds?

The closest result we found is in Bourbaki (Groupes et algebres de Lie, chapitre 7, §3, Proposition 4), stating: let $\mathfrak{g}$ be a Lie algebra of rank $l$ over a field of characteristic $0$. Let $c$ be the nilpotency class of its Cartan subalgebras, and $X\in \mathfrak{g}$. There exists a subalgebra of dimension $l$ with nilpotency class at most $c$ containing $X$.

It is worth mentioning a result about regular elements. An element is called regular if its centralizer has dimension equal to the rank of $\mathfrak{g}$. In that case, this centralizer is a Cartan subalgebra, and every Cartan subalgebra arises like this.