The triple linking number of an oriented surface link was defined as an
analogical notion of the linking number of a classical link. We consider a
certain $m$-component $T^2$-link ($m \geq 3$) determined from two commutative
pure $m$-braids $a$ and $b$. We present the triple linking number of such a
$T^2$-link, by using the linking numbers of the closures of $a$ and $b$. This
gives a lower bound of the triple point number. In some cases, we can determine
the triple point number, which is a multiple of four.