Proof that a set of matrices form a group. Is it a Lie group? Calculate the generators

Question

I have to proof that the set of matrices that verify this equation form a group.In that case, iis it a lie Group? Which is the dimension? Note that the we are working in the complex space C in 4 dimensions.

So far i have found that this group of matrices is called Complex Lorentz Group.(Is it a subgroup of the pseudo-unitary group(3,1). In this case we are working with a signature diag(-1,1,1,1)

How can if find the generators of the group?Dimensions of the generators?

And the last question,is there any subgroups such a cartan subalgebra or casimir opreartor

Image of the problem:

Answer 1

Hint: A matrix $X$ will be an element of the Lie Algebra associated with the Lie Group $G$ if for all $t \in \Bbb R$, we have $\exp(tX) \in G$ (or $\exp(itX) \in G$ if you are using the physics convention).

Note that $U\exp(tX)U^\dagger = \exp(t \,UXU^\dagger)$.