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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?

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1 Answer 1

up vote 2 down vote accepted
+50

For $SO(2,2)$, You should check that it preserves the quadratic form $-x^2-y^2+z^2+w^2$, and that the matrix has determinant one. I guess the $+$ means that the transformation preserves both orientations, meaning that in you example both smaller determinants have to be equal to one.

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