I am a little confused as to how to compute generally the Lie algebra of a Lie group and viceversa, namely the Lie groups (up to diffeomorphism) having a certain Lie algebra.
The way I did this for classical groups such as $O(n)$ or $SL_n(\mathbb{R})$ was to express them as fibres over a regular value of a smooth map and then explicitly computing the tangent space at the identity as the kernel of the differential. This method however works only in very specific cases.
- How does one, for instance, compute the Lie algebra of the group $SO(2) \bar{\times} \mathbb{R}^4$ (by $\bar{\times}$ I mean the semi-direct product).
- Which connected Lie groups up to diffeomorphism have the following Lie algebra
$\left\{\left(\begin{array}{ccc} x & y & w \\ z & -x & v \\ 0 & 0 & 0 \end{array} \right), \qquad x,y,z,v,w\in \mathbb{R}\right\}?$
Thanks in advance for any help.