10. There are six thin sticks, among which the lengths of the two longer ones are $\sqrt{3} a$ and $\sqrt{2} a$, and the lengths of the other four are all $a$. Using them to form a triangular pyramid. Then the cosine value of the angle between the two longer edges is $\qquad$
$10 \cdot \frac{\sqrt{6}}{3}$.
As shown in Figure 4.
At first glance, it seems that two diagrams, Figure 4(a) and Figure 4(b), can be drawn, but if we reconsider the existence of the two diagrams, we immediately find that the figure is actually a trap, and Figure 4(a) does not exist at all. Because by taking the midpoint $E$ of $A C$ and connecting $B E$ and $E D$, we get $B E=E D=\frac{\sqrt{2}}{2} a$. Thus, $B E+E D=\sqrt{2} a<\sqrt{3} a$ $=B D$, which contradicts the triangle inequality that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, Figure 4(a) should be excluded.

In Figure 4(b), the cosine of the angle between the two longer edges is $\cos \angle B D C=\frac{\sqrt{2}}{\sqrt{3}}=\frac{\sqrt{6}}{3}$.