If $a=0$, the equation obviously holds. Therefore, we can assume $a, b, c$ are positive integers,
$$a=p_{1}^{q_{1}} \cdots p_{s}^{\sigma_{3}}, \quad b=p_{1}^{\beta_{1}} \cdots p_{3}^{\beta_{3}}, \quad c=p_{1}^{\gamma_{1}} \cdots p_{3}^{\gamma_{3}}.$$

By Corollary 4, we get
$$\begin{array}{l} (a,[b, c])=p_{1}^{\eta_{1}} \cdots p_{p_{j}^{\eta_{j}}}, \\ \eta_{j}=\min \left(\alpha_{j}, \max \left(\beta_{j}, \gamma_{j}\right)\right), \quad 1 \leqslant j \leqslant s \\ {[(a, b),(a, c)]=p_{1}^{\tau_{1}} \cdots p_{s}^{\tau_{s}},} \\ \tau_{j}=\max \left(\min \left(\alpha_{j}, \beta_{j}\right), \min \left(\alpha_{j}, \gamma_{j}\right)\right), \quad 1 \leqslant j \leqslant s \end{array}$$

It is easy to verify that, regardless of the size relationship between $\alpha_{j}, \beta_{j}, \gamma_{j}$, we always have $\tau_{j}=\eta_{j}(1 \leqslant j \leqslant s)$. This proves the desired conclusion. It is relatively difficult to prove this relationship directly using the method of §4.