I'm absolute novice with Lie groups and Lie algebras topic and I have a problem just with starting with a task I got.
The problem is to find Lie algebra of a subgroup $G$ of the $SL(2,\mathbb{C})$, which is of the form:
$\begin{align} G=\{g=t \sigma_{0}+x\sigma_{1}+y\sigma_{2}+iz\sigma_{3}\}, with: t^{2}-x^{2}-y^{2}+z^{2}=1,\end{align}$
where: $x,y,z,t\in\mathbb{R}$ and $\sigma_{i}$ are Pauli matrices (including identity with i=0). There is also a hint for this exercise, which says that elements of $G$ fulfill the equation
$g^{\dagger}Dg=D$
for some diagonal matrix $D$.
It's not hard to figure out that Lie algebra of $SL(2,\mathbb{C})$ must consist of matrices with $0$ trace. But I have no idea how to use properties of subgroup $G$ to get more restrictions on Lie algebra of it. I have tried to write down that something like this:
$\forall_{r\in\mathbb{R}}\ \exp(rA) \in\ G\implies\ \exp(rA)^{\dagger}\ D\ \exp(rA)= \exp(rA^{\dagger})\ exp(rA)\ D=D$
and therefore
$exp(rA^{\dagger})\ exp(rA)=\mathbb{1}$,
so this would mean that if $A$ is normal, then it implies its antihermiticity. Unfortunately I do not know how to justify that it should be normal.
When I try to find some matrix that $\forall_{r\in\mathbb{R}}\ \exp(A\ r)\in G$ with brutforce calculations, I got very ugly results and I believe I'm missing something that might simplify those.
I will be grateful for any help with how to start with this exercise!