I was just curious if anyone knows what kind of constraints one can place on $SL(4,R)$ the set of 4x4 invertible matrices with unit determinant to obtain: $SL(2,C)$ the set of 2x2 invertible complex matrices with unit determinant. I'm guessing that if one places a particular constaint on the former set then they are isomorphic to the latter?
Essentially I'm trying to show that a particular set of matrices I have might be represented in terms of $SL(2,C)$ rather than the current form I have them in $SL(4,R)$. Thanks a ton!
If $\Lambda$ is of the form: $$\Lambda = \begin{bmatrix} a&b&c&d \\ -b&a&-d&c \\ e&f&g&h \\ -f&e&-h&g \end{bmatrix}$$ then we can isomorphically map it to: \begin{bmatrix} a+ib&c+id \\ e+if&g+ih \end{bmatrix}