We consider a class of quasigroup identities (with one operation symbol) of the form $x_1x_2\cdot x_3x_4=x_5x_6\cdot x_7x_8$ and with $x_i\in\{x,y,u,v\}$ ($1\leq i\leq8)$ with each of the variables occurring exactly twice in the identity. There are 105 such identities. They generate 26 quasigroup varieties. The lattice of these varieties is given.