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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?

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up vote 2 down vote accepted

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.

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