19. (15 points) As shown in Figure 3, given a cube $A B C D$ $A_{1} B_{1} C_{1} D_{1}$ with an edge length of 2, point $E$ is the midpoint of edge $C D$. Find the distance between the skew lines $A_{1} C_{1}$ and $B_{1} E$.
19. As shown in Figure 7, establish a spatial rectangular coordinate system $D-xyz$,
then $A_{1}(2,0,2), C_{1}(0,2,2), B_{1}(2,2,2), E(0,1,0)$.
$$
\begin{array}{l}
A_{1} C_{1}=(-2,2,0), \\
B_{1} E=(-2,-1,-2), \\
A_{1} B_{1}=(0,2,0) .
\end{array}
$$

Let $PQ$ be the common perpendicular segment of the skew lines $A_{1} C_{1}$ and $B_{1} E$, and
$$
\begin{array}{l}
A_{1} P=\lambda A_{1} C_{1}=(-2 \lambda, 2 \lambda, 0), \\
B_{1} \boldsymbol{Q}=\mu B_{1} E=(-2 \mu,-\mu,-2 \mu) .
\end{array}
$$

Then $PQ=PA_{1}+A_{1}B_{1}+B_{1}\boldsymbol{Q}$
$$
=(2 \lambda-2 \mu,-2 \lambda-\mu+2,-2 \mu) \text {. }
$$

Since $PQ \perp A_{1} C_{1}, PQ \perp B_{1} E$, we have
$$
\begin{array}{l}
P \boldsymbol{P} \cdot A_{1} \boldsymbol{C}_{1}=0, PQ \cdot B_{1} E=0 . \\
\text { Also, } PQ \cdot A_{1} C_{1}=-8 \lambda+2 \mu+4, \\
PQ \cdot B_{1} E=-2 \lambda+9 \mu-2, \\
\text { then }\left\{\begin{array}{l}
-8 \lambda+2 \mu+4=0, \\
-2 \lambda+9 \mu-2=0 .
\end{array}\right.
\end{array}
$$

Solving these, we get $\lambda=\frac{10}{17}, \mu=\frac{6}{17}$.
Therefore, $\boldsymbol{PQ}=\left(\frac{8}{17}, \frac{8}{17},-\frac{2}{17}\right)$,
$$
|PQ|=\sqrt{\left(\frac{8}{17}\right)^{2}+\left(\frac{8}{17}\right)^{2}+\left(-\frac{12}{17}\right)^{2}}=\frac{4 \sqrt{17}}{17} .
$$