Working through homework and I run into this problem:
Suppose the Lie group $SO^{+}(2,2)$ is presented as the group of all transformations in its associated space. How do you determine whether a given 4x4 transformation matrix is an element of the group?
I'm slightly familiar with the $SO^{+}(1,3)$ Lie group, but have almost no experience with group theory as a whole. So far, I seem to remember the conditions require orthogonality between coordinates, preservation of the orientation (of subgroups?), and elements with a determinant of 1, but still feel like I'm missing something. I expect given a matrix representation of the form:
$\left( \begin{array}{ccc} a & b & 0 & 0\\ c & d & 0 & 0\\ 0 & 0 & e & f\\ 0 & 0 & g & h\end{array} \right) $
Do I need to ensure the determinants of the components
$\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) $ and $\left( \begin{array}{ccc} e & f \\ g & h \\ \end{array} \right) $
are related in a certain way to prevent orientation shifts (such as reflections), or do I only need to pay attention to the total determinant of a transformation?