(10) As shown in the figure, at first glance, two diagrams can be made, as shown in Figure 1 and Figure 2. However, if we slightly consider the existence of the two diagrams, we immediately find that Figure 1 does not exist. By taking the midpoint $E$ of $AC$ and connecting $BE$ and $ED$, we get $BE = ED = \frac{\sqrt{2}}{2} a$, thus
$$
BE + ED = \sqrt{2} a < \sqrt{3} a = BD,
$$

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 1 should be excluded. In Figure 2, the cosine of the angle between the two longer edges is $\cos \angle BDC = \frac{\sqrt{2}}{\sqrt{3}} = \frac{\sqrt{6}}{3}$.
10 There are six thin sticks, among which the two longer ones are $\sqrt{3} a$ and $\sqrt{2} a$, and the other four are all $a$. Using them to form a triangular pyramid, the cosine value of the angle between the two longer edges is