4・154 From the system of equations
$$\left\{\begin{array}{l}
a=\cos u+\cos v+\cos w \\
b=\sin u+\sin v+\sin w \\
c=\cos 2 u+\cos 2 v+\cos w \\
d=\sin 2 u+\sin 2 v+\sin 2 w
\end{array}\right.$$

eliminate \( u, v, w \).
[Solution] From the system of equations, we have
$$\begin{array}{l}
a^{2}+b^{2}=3+2[\cos (u-v)+\cos (v-w)+\cos (w-u)], \\
a^{2}-b^{2}=C+2[\cos (u+v)+\cos (v+w)+\cos (w+u)], \\
2 a b=d+2[\sin (u+v)+\sin (v+w)+\sin (w+u)],
\end{array}$$

Therefore, $\left(\frac{a^{2}-b^{2}-c}{2}\right)^{2}+\left(\frac{2 a b-d}{2}\right)^{2}$
$$\begin{array}{l}
=3+2[\cos (u-v)+\cos (v-w)+\cos (w-u)] \\
=a^{2}+b^{2}
\end{array}$$

That is, $\left(a^{2}-b^{2}-c\right)^{2}+(2 a b-d)^{2}=4\left(a^{2}+b^{2}\right)$.