Sure, the supply lines can run through the toekick. But, the supply lines aren't the problem when moving plumbing. The drain lines are the bigger issue. You will have to rework the drain piping in the basement (or jackhammer the slab and rework it). The drain lines take up a lot of space. The piping is 2 1/2" and won't fit in a 2"x4" wall. It needs a 2"x6" pony wall to fit into.

Moving plumbing needs a licensed plumber, not a "contractor". It will also probably require that your licensed plumber pull a permit from your municipality and be inspected.

hi elizahu,

The P trap is the bendy thing where some of the drain water remains at all time; water gets trapped while it also lets new drain water flow out. In 99% of kitchens you will find it right under your sink drain hole. Look at the pipe coming out of it. It's approximately 1.5" diameter.

This big pipe *can* be sent through the cabinets. This is why Ikea Blum drawers only go 21" deep. It leaves room for the big drain pipe.

So this is how you get the drain to be "invisible" in a small space kitchen.

--

The other subject that must be studied is the length of the proposed new pipe. The total length of this entire segment must be known before anyone will agree that "it can be done" because this segment has to be shorter than the allowed limit. The segment of pipe in question is the "Trap ARM" in case you want to learn more by web searching before you consult a Master Plumber or your local authorities.

Hth

I forgot to answer your question with a blunt answer. Here goes.
"Can the plumbing pipe go under the cabinet where kickboard is?"

No. The drain pipe might fit by going through holes in the cabinet sides, and these holes have to be at exactly the right height. Knowing this you may now re-read the above post.

Assuming that the peninsula is permanently installed running the lines under the cabinet would afford the lines the same level of protection as if they were run through a chase and surely more protection than if it was run through the cabinets, however the actual distance you could run would be dependant upon how the cabinets are made.

Typically the vertical clearance in the underside of the cabinet base is the same as the vertical height of the toe kick, which is generally 3-1/2"

Under the IRC a kitchen sink drain line is 1-1/2" pipe and the outside diameter of 1-1/2" sched.40 PVC is 1-7/8.

Subtracting 1-7/8" from 3-1/2" leaves us 1-5/8" for pipe pitch, (assuming of course that the downstream end of the pipe is flat with the floor.)

Code requires a 1/4" per/ft pitch so the absolute maximum horizontal length we could fit in the space is 6=1/2'of pipe, however that is assuming that on the downstream end of the pipe you dig into the concrete and set a Tee so that the pipe will be flush with the floor on the downstream end and that you make the hole in the cabinet base on the upstream end large enough to allow the pipe to be flush with the underside of the cabinet base, otherwise you will loose about 1/4" to the sidewall of the 1/4 bend turning up, which would be equal to 1' of run.

The problem is venting the trap. Under the IRC for a fixture arm the bottom of the pipe at the trap weir may not be higher than the top of the pipe at the vent opening therefore you would need to provide a vent near the sink.

The IRC allows an unlimited number of AAV's (Air Admittance Valves commonly called Studor vents or cheater vents) You could then connect a vertical riser from the 1/4bend at the cabinet base and run a vertical up to the fixture arm height where you install a Tee, then extend the vertical riser at least 6" and install an AAV.

The preferred option would be to install an island loop vent in the cabinet at the sink location.

Under the UPC they require the kitchen drain line to be 2" pipe therefore you could configure in about the same manner but the maximum horizontal length that would fit under the cabinet is about 4'.

Sorry, I forgot something else.
re:
"Can the plumbing pipe go under the cabinet where kickboard is?"

It IS possible but it involves rebuilding a lot more of the whole house drain plumbing system.