Circuit of Instrumentation Amplifier
In short, with the split and grounded R1 circuit, A1 and A2 amplify both common and differential mode signals by the same gain.







simulate this circuit &ndash; Schematic created using CircuitLab

So their outputs will be easily saturated when trying to extract a tiny differential mode signal superposed on a high common mode.



On the other hand the "classical" floating R1 circuit always amplifies common mode voltages by unity gain, whatever differential mode  gain you set using R1/R2.







simulate this circuit



That's the clever thing about it, it helps extracting "small" differential on "high" common mode, improves CMMR and dynamic range.