obviously, unitary matrices with determinant=1

Generators are just useful way to look at the group.

I read some more of the book and after going over it several times, I figured out that equation. Difficult to actually work out the actual matrices since they require an infinite sum. But I assume that it’s been done to a fairly high accuracy. Maybe that’s later in the book.

SLD

I suspect you are doing all this as a hobby and skipping a lot of prerequisites, right?

Computing exponent of the matrix is actually not that difficult and is done with absolute precision in fixed amount of steps but it's rarely done in practice.