I pull the basket/pot out. For lilies or any plant where the top will get damaged I cut off all the stems/leaves. Maybe 4-6" above the soil.

I get the root ball out of the pot, if I have to cut the pot away I do that. I rinse off the root ball to see what I can see.

I use a heavy knife (doesn't need to be sharp) to cut the ball up, or a shovel if it's a big dense ball. Maybe in half then I can see more rhizomes. I'm try to find the biggest best looking rhizomes. Keep rinsing and cutting until I have some rhizomes chunks.

You'll see how roots go down from the rhizome, the leaves/stems to up and about how deep the rhizome was planted, remember to place the rhizome the same way in the new pot.

I plant the best rhizomes into the new pot. I use clay soil from my yard but their are other choices like rock dust, pea gravel, kitty litter, aquaponic media, etc. I don't think it matters much myself.

I break a Jobe's Fruit Tree fertilizer spike into 3 or 4 chunks and push 3 or 4 into the soil around the edge.
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I soak the pot in water for maybe 30 minutes, or just keep adding water until the soil is soaked to reduce the amount of "dust" that will get into the pond water when I put the pot back.

I find lilies are pretty forgiving about being divided. Just pull them out of the pots (or cut the pot off - if they are really overgrown they might be wedged in there pretty good), rinse off any remaining dirt or gravel and start separating.You should see distinctive tubers - cut them apart. Any tuber that has a growing tip (a point where new growth us emerging) can be potted up and will grow into a new plant.

You will probably find you have far more tubers than you need for your pond. That's a nice problem to have!

Fantastic! Thank you. From what I read I think I am splitting these plants the same way I split my Iris in the garden.

Lilies: Do I cut them right back before I split and transplant?

Lily pads: What do I do with these?

And most importantly, when do I do this?

Any time you like.

Fall is a good time imo just after the first frost because you have to deal with the dead leaves anyways. So two birds. In this case I store the rhizomes for winter when I live in a freezing climate like NY. Lots of web sites with how to info on that, lots of different ways. In milder climates like San Jose CA and Phoenix I pull the pots out and leave them until spring when I divide. Or in bogs I just lower the water level until spring.

Depends on the kind of plant but some, like Canna, can't handle being in water until after they've sprouted at least some green. So I pot them in spring and water like any plant. When I get good growth I put them into the pond or bog. All marginals I've run into so far seem to prefer this treatment. Otherwise rhizomes tend to rot.

In recent times I've opted to not have any plants inside the pond and instead just use bogs and bog type beds around the pond. A deal like this:
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Makes life easier and keeps the pond cleaner. I can lower the pond water level a inch or so and the plants go from being in the pond to not in the pond. Then I can divide whenever I like.