This question is a follow-up to this post, from which I quote:

Let $\mathfrak{e}$ be the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is isomorphic to the Lie algebra of the group of isometries of the plane. Its central extension $\tilde{\mathfrak{e}}$ is defined as the 4-dimensional Lie algebra defined by adding a central generator $Z$ and the additional nonzero bracket $[X,Y]=Z$.

It seems to me that since $\mathrm{exp}$ is a diffeomorphism on a Carnot group then the latter's Lie algebra $\mathfrak{g}$, should not contain either $\mathfrak{e}$ or $\tilde{\mathfrak{e}}$. However, where can I find a proof of the fact that no Carnot group contains either $\mathfrak{e}$ or $\tilde{\mathfrak{e}}$ (without going through the Sato-Dixmier result of the quoted post).

This question is a follow-up to this post, from which I quote:

Let $\mathfrak{e}$ be the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is isomorphic to the Lie algebra of the group of isometries of the plane. Its central extension $\tilde{\mathfrak{e}}$ is defined as the 4-dimensional Lie algebra defined by adding a central generator $Z$ and the additional nonzero bracket $[X,Y]=Z$.

It seems to me that since $\mathrm{exp}$ is a diffeomorphism on a Carnot group then the latter's Lie algebra $\mathfrak{g}$, should not contain either $\mathfrak{e}$ or $\tilde{\mathfrak{e}}$. However, where can I find a proof of the fact that no Carnot group contains either $\mathfrak{e}$ or $\tilde{\mathfrak{e}}$ (without going through the Sato-Dixmier result of the quoted post).

This question is a follow-up to this post, from which I quote:

Let $\mathfrak{e}$ be the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is isomorphic to the Lie algebra of the group of isometries of the plane. Its central extension $\tilde{\mathfrak{e}}$ is defined as the 4-dimensional Lie algebra defined by adding a central generator $Z$ and the additional nonzero bracket $[X,Y]=Z$.

It seems to me that since $exp$ is a diffeomorphism on a Carnot group then the latter's Lie algebra $\mathfrak{g}$, should not contain either $\mathfrak{e}$ or $\tilde{\mathfrak{e}}$. However, where can I find a proof of the fact that no Carnot group contains either $\mathfrak{e}$ or $\tilde{\mathfrak{e}}$ (without going through the Sato-Dixmier result of the quoted post).

This question is a follow-up to this post, from which I quote:

Let $\mathfrak{e}$ be the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is isomorphic to the Lie algebra of the group of isometries of the plane. Its central extension $\tilde{\mathfrak{e}}$ is defined as the 4-dimensional Lie algebra defined by adding a central generator $Z$ and the additional nonzero bracket $[X,Y]=Z$.

It seems to me that since $\mathrm{exp}$ is a diffeomorphism on a Carnot group then the latter's Lie algebra $\mathfrak{g}$, should not contain either $\mathfrak{e}$ or $\tilde{\mathfrak{e}}$. However, where can I find a proof of the fact that no Carnot group contains either $\mathfrak{e}$ or $\tilde{\mathfrak{e}}$ (without going through the Sato-Dixmier result of the quoted post).