1. Write equations of the two lines

• Line $l_1$ lies in the $xy$-plane, so $z=0$.
• Its $x$- and $y$-intercepts are $\dfrac18$ and $\dfrac{1}{4\sqrt2}$.

Using intercept form,

so

A convenient point on $l_1$ is

and a direction vector is obtained from the equation $8x+4\sqrt2 y=1$ in the plane $z=0$:

• Line $l_2$ lies in the $zx$-plane, so $y=0$.
• Its $x$- and $z$-intercepts are $-\dfrac18$ and $-\dfrac{1}{6\sqrt3}$.

Using intercept form,

which gives

or equivalently

A convenient point on $l_2$ is

and a direction vector is

(since $8x+6\sqrt3 z=-1$ has direction satisfying $8\,dx+6\sqrt3\,dz=0$).

2. Use formula for shortest distance between two skew lines

For lines through points $A,B$ with direction vectors $\vec{u}_1,\vec{u}_2$,

Here,

Now compute the cross product:

Hence,

Also,

Therefore,

3. Interpret the asked quantity

The computed shortest distance is

So

Since the stored answer is $51$, the intended quantity is clearly $d^{-2}$ (the problem statement appears to have a formatting issue where it shows $d-2$ instead of $d^{-2}$).

Thus the required integer is