Here are the photos of the other brackets. The white one is the Island Support System by countertopbracket.com. http://www.countertopbracket.com/ProductDetails.asp?ProductCode=ISS These come in a choice of white or black, I got white to match the cabinets.
The black long ones are the Hidden Island Support Bracket, supported by a 1x3 in front, and the 2x6 and 2x8 in the back of the cabinet. by originalgranitebracket.com. https://originalgranitebracket.com/collections/hidden-cabinet-brackets/products/hidden-island-support-bracket These only come in black, I painted the underside white so they won't really show even if someone is sitting on the couch and looking under the counter.
All of these use cantilever principle, the heavy countertop on the cabinet section of the metal bar prevents the overhang section of the bar from being pushed down.
As you can see in the top photo, I reinforced the end cabinet with a vertical 2x4 in each corner, supporting the 2x6 across the top. Again that's probably overkill, but it only cost a few dollars and it gives me peace of mind with a 15 inch overhang on two sides.
Below you can see how I had to notch the 2x6 quite a bit to fit the corner bracket on there.
Below you can see how all the drawer slides fit with a 2x4 behind them.
The photo below shows how the brackets screw into the 1x3, and are supported by the 2x8 and 2x6.
The photo below shows the 2x8 in the back of the cabinet, and the Maximera drawer slides have just enough clearance. You can also see a vertical 1x4 below the 2x8, which I had to use instead of a 2x4 because the waste baskets would not fit otherwise. The 1x4 is used to give the silver brackets from the adjacent cabinet something to bite into. I don't trust particleboard for structural fasteners, but particleboard works great when sandwiched between a bracket (or screwhead) and a piece of lumber.
Q