1. On the surface of a cube, there exist ascending lines on the diagonals of the square faces. If the distance between two of these lines is 1, then the volume of the cube is ( ).
(A) 1
(B) $3 \sqrt{3}$
(C) 1 or $3 \sqrt{3}$
(D) $3 \sqrt{3}$ or $3 \sqrt{2}$
$-1 .(\mathrm{C})$.
Let the edge length of the cube be $x$. If the distance between the face diagonal $A C$ and $B^{\prime} D^{\prime}$ is 1, then $x=1$, and thus the volume $V$ $=x^{3}=1$.

If the distance between the face diagonal $A C$ and $B C^{\prime}$ is 1, then $\frac{\sqrt{3}}{3} x=$ $1, x=\sqrt{3}, V=x^{3}=3 \sqrt{3}$.
Thus, the volume of the cube is 1 or $3 \sqrt{3}$.