Lie-Trotter Decomposition and Leapfrog Method Splitting Method

Question

I'm not sure I can ask this question, if it causes some problem, then I would immediately delete the post.

I'm looking at "Spinsim" Package, and at the guide pdf, it shows some math technique:

I'm not so good in mathematics, and I have just found out about Lie-Trotter Decomposition in the pdf.

I just don't get how the above $T$ becomes the below $T$ (by leapfrog splitting method, they say)

I tried to read the article in appendix [22]. I think it is related to this:

I just don't get how the arctan terms, and $w_x$ squared terms appear just by that. It says about commute relations, but I have no idea what it means since the spin operators do not commute.

Thanks for the help.