Example 3 (Hainan 2008) A certain edge of a geometric body has a length of $\sqrt{7}$. In the front view of the geometric body, the projection of this edge is a line segment with a length of $\sqrt{6}$. In the side view and top view of the geometric body, the projections of this edge are line segments with lengths of $a$ and $b$ respectively. Then the maximum value of $a+b$ is ( )
A. $2 \sqrt{2}$
B. $2 \sqrt{3}$
C. 4
D. $2 \sqrt{5}$
Let the edge length of the geometric body be $\sqrt{7}$ for the edge $A C_{1}$. Construct a rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$ with $A C_{1}$ as the diagonal. Then the line segments $D C_{1}, B C_{1}$, and $A C$ are the projections of the edge $A C_{1}$ in the front view, side view, and top view of the geometric body, respectively. Let $A B=x, A D=y, A A_{1}=z$, then
$$\left\{\begin{array}{l}
x^{2}+y^{2}=b^{2}, \\
y^{2}+z^{2}=a^{2}, \\
z^{2}+x^{2}=6, \\
x^{2}+y^{2}+z^{2}=7,
\end{array}\right.$$

Thus, we get $a^{2}+b^{2}=8$.
By the Cauchy-Schwarz inequality, we have $(a+b)^{2} \leqslant 2\left(a^{2}+b^{2}\right)=16$,
so $a+b \leqslant 4$, with equality if and only if $a=b=2$. Therefore, the answer is C.