Dr. Mitchell’s Physics talks about advanced level Classical Mechanics, and this chapter about Poisson bracket and canonical transformation is quite fresh in the sense not many resources are available.

when H is implicit time dependent, not explicit time-dependent, which can be illustrated with an example,

Some properties of Poisson Bracket

Then what is canonical transformation? only when PB is preserved. Or, we say canonical transformation preserves all Poisson brackets. A example below can illustrate clearly what that means.

when we say “commute under the Poisson bracket”, is it equivalent to invariant under the Poisson bracket”?