  Motivated by works on extension sets in standard domains we introduce a
notion of the Carath\'eodory set that seems better suited for the methods used
in proofs of results on characterization of extension sets. A special stress is
put on a class of two dimensional submanifolds in the tridisc which not only
turns out to be Carath\'eodory but also provides examples of two dimensional
domains for which the celebrated Lempert Theorem holds. Additionally, a
recently introduced notion of universal sets for the Carath\'eodory extremal
problem is studied and new results on domains admitting (no) finite universal
sets are given.
