The Diamond Splitter
Because the rail splitter sources and sinks the currents in the O/P stages, the "SQ" of the splitter is almost as important as the SQ of the buffers themselves. Thus, a simple opamp-based splitter is not good enough to complement the O/P buffers. So, what do we do?
The first step is to review a few rail splitter configurations. A really good resource for designing rail splitters is Tangent's Virtual Ground Circuits. Two basic rail splitters can be made from an off-the-shelf TLE device. In this case the TLE2426 is the right one because the V+ is 24V.
Like this:

Simple Rail Splitters
The TLE is very good as a low current virtual ground circuit, but its maximum current is 20mA. Since our buffers are each running at 20mA and could possibly draw more than 40mA from the splitter we need a higher current output virtual ground. A common way to do this is to use a TLE device as only a reference for a high current buffer as shown in the right side drawing. There are many types of buffers that can be used here including single chip buffers (BUF634), high current opamps in unity gain mode, and even discrete buffers. For this amp we want to use a discrete buffer similar to the O/P buffers. The splitter butter, therefore, should have a similar a topology as we can make. In other words the splitter buffer should look like a diamond buffer. Like this:

Diamond Buffer for the Splitter
The only thing that's different from the O/P buffers is that we don't need CCSs on the emitters and we have paralleled O/P transistors to handle the currents from both O/P buffers at the same time. All we need now is the feedback circuit that will hold the VG of the diamond splitter. To do this effectively we need lots of gain, such as, an opamp. In this case we want to include the buffer in the feedback loop of the opamp so that the opamp will use all of its gain to hold the buffer's output to the reference voltage. And for simplicity we use the TLE device to provide the reference at half the rail voltage.

The Diamond Splitter
The resistor between the TLE and the opamp is a gate stopper to prevent oscillation. The opamp feedback capacitor is there to shunt high frequencies around the opamp to create unity gain at high frequencies. Again to prevent oscillation. Otherwise the opamp is allowed to use all of its gain to maintain the VG reference as the transistors source and sink current.
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