The Lie algebra of a group of matrices

Question

How does one find the Lie Algebra of a Lie group G which is given by matrices

$$\left(\begin{array}{cccc} \cos \theta & -\sin \theta & x & y \\ \sin \theta & \cos \theta & z & w \\ 0 & 0 & \cos \theta & -\sin \theta \\ 0 & 0 & \sin \theta & \cos \theta \end{array} \right)?$$

From the shape of the matrices, it seems G is isomorphic to the semidirect product of $SO(2)$ with $\mathbb{R}^4$.

I read in previous posts that one might try to identify which matrices exponentiate to matrices of this form, although this seems complicated. Is there another approach to tackle the problem?

Answer 1

You can see which matrices exponentiate to matrices of this form. For example if $A \in \mathfrak o(2)$, i.e. $A$ is a skew-symmetric $2\times 2$ matrix then the block matrices $$ \left(\begin{array}{c|c} A & 0 \\\hline 0 & 0 \end{array}\right), ~~ \left(\begin{array}{c|c} 0 & 0 \\\hline 0 & A \end{array}\right) $$ exponentiate to $$ \left(\begin{array}{c|c} e^A & 0 \\\hline 0 & Id \end{array}\right), ~~ \left(\begin{array}{c|c} Id & 0 \\\hline 0 & e^A \end{array}\right) $$ which are elements of your group since $e^A \in SO(2)$. Similarly, for any $2\times 2$ matrix $B$, $$ \left(\begin{array}{c|c} 0 & B \\\hline 0 & 0 \end{array}\right) $$ exponentiates to $$ \left(\begin{array}{c|c} Id & B \\\hline 0 & Id \end{array}\right) \in G $$ since it is nilpotent. By dimensionality, the span of these must be the entire Lie algebra.