I Frobenius theorem applied to frame fields

[fresh_42]

Sorry, but the above Borel subgroup of $\operatorname{SL}(2,\mathbb R)$ is not itself a Lie group ?

It is of course a Lie group. It's a two-dimensional smooth manifold. A typical element looks like
$$
\begin{pmatrix}e^t&c\\0&e^{-t}\end{pmatrix}
$$
The toral element, the diagonal matrix is a flow through time, and the unipotent element makes it non-abelian. The group is
$$
\biggl\langle \begin{pmatrix}e^t&c\\0&e^{-t}\end{pmatrix} \biggr\rangle =\left\{\left.\begin{pmatrix}x&y\\0&z\end{pmatrix}\,\right|\,x\cdot z= 1\right\} .
$$