  We prove new estimates of the $K$-divisibility constants for some special
Banach couples. In particular, we prove that the $K$-divisibility constant for
a couple of the form $(U\oplus V, U)$ where $U$ and $V$ are non-trivial Hilbert
spaces equals $2/\sqrt{3}$. We also prove estimates for the $K$-divisibility
constant of the two-dimensional version of the couple $(L_2,L_\infty)$, proving
in particular that this couple is not exactly $K$-divisible. There are also
several auxiliary results, including some estimates for relative Calder\'on
constants for finite dimensional couples.
