  We prove the $K$-polystability of all smooth complex Fano threefolds
admitting an effective action of $\text{SL}_2$ but not of a 2-torus or 3-torus.
In particular, the existence of K\"{a}hler-Einstein metrics on varieties in the
families (1.10), (1.15), (1.16), (1.17), (2.21), (2.27), (2.32), (3.13),
(3.17), (3.25) and (4.6) of the Mori-Mukai classification of smooth Fano
threefolds is proved.
