Calculations in conformal theory needed for verifying the Alday–Gaiotto–Tachikawa hypothesis

Explicitly verifying the Alday–Gaiotto–Tachikawa (AGT) relation between the conformal blocks controlled by the $W_N$ symmetry and $U(N)$ Nekrasov functions requires knowing the Shapovalov matrix and various triple correlators for $W$-algebra descendants. We collect the simplest expressions of this type for $N=3$ and for the two lowest descendant levels together with the detailed derivations, which can now be computerized and used in more general studies of conformal blocks and AGT relations at higher levels.