Example 5 Find positive integers $a, b, c$, such that
$$
[a, b, c]=a+b+c .{ }^{[2]}
$$
(2005, Austrian Mathematical Olympiad)
Let's assume $a \leqslant b \leqslant c$, where the equality cannot hold simultaneously.
Therefore, $a+b<2 c \Rightarrow c<a+b+c<3 c$.
Combining $l c=[a, b, c]=a+b+c$, we get $l=2$, i.e., $a+b=c$.
On the other hand, $k b=[a, b, c]=a+b+c=2 a+2 b$. Hence $b \leqslant 2 a$.
Combining $2 a \leqslant 2 b$, we get $b=a$ or $b=2 a$.
(1) When $b=a$, $c=a+b=2 a$.
Then $[a, a, 2 a]=2 a \neq a+b+c$, the equation has no solution.
(2) When $b=2 a$, $c=3 a$.
Then $[a, 2 a, 3 a]=6 a=a+b+c$, the equation always holds.