Computing Lie algebra of a subgroup

Question

I will like to know how does one compute the Lie algebra of an abstractly given subgroup of a Lie group? Specifically, let $G = \mathrm{SO} ( n + 1, 1 )$ and consider the flow $$ g_t = \begin{pmatrix} \cosh t &0 &- \sinh t\\ 0 &I_n &0\\ - \sinh t &0 &\cosh t \end{pmatrix} $$ and the associated expanding subgroup $H = \{ h \in G \mid g_{-t} h g_t \to e \textrm{ as } t \to \infty \}$. I want to understand how to get the Lie algebra of $H$ though I do know the final answer.

Answer 1

$\mathfrak{h}=\lbrace \xi \in \mathfrak{g}, \frac{d}{ds}\vert_{s=0}(g_{-t}e^{s\xi}g_t) \to 0 $ as $t \to \infty \rbrace $

So you make the computation, gives you a condition, and you check who realizes it in $\mathfrak{g}$...

Does it help?