Simplify: $\;$ $\left(2 - \dfrac{a}{4} - \dfrac{4}{a}\right) \left[\left(a - 4\right) \sqrt[3]{\left(a - 4\right)^{-3}} - \dfrac{\left(a^2 - 16\right)^{\frac{-1}{2}} \left(a - 4\right)^{\frac{-1}{2}}}{\left(a + 4\right)^{\frac{-3}{2}}}\right] \left(\dfrac{a + 4}{a - 4}\right)$
$\left(2 - \dfrac{a}{4} - \dfrac{4}{a}\right) \left[\left(a - 4\right) \sqrt[3]{\left(a - 4\right)^{-3}} - \dfrac{\left(a^2 - 16\right)^{\frac{-1}{2}} \left(a - 4\right)^{\frac{-1}{2}}}{\left(a + 4\right)^{\frac{-3}{2}}}\right] \left(\dfrac{a + 4}{a - 4}\right)$
$= \left(\dfrac{8a - a^2 - 16}{4a}\right) \left[\left(a - 4\right) \left(a - 4\right)^{-3 \times \frac{1}{3}} - \dfrac{\left(a + 4\right)^{\frac{-1}{2}} \left(a - 4\right)^{\frac{-1}{2}} \left(a - 4\right)^{\frac{-1}{2}}}{\left(a + 4\right)^{\frac{-3}{2}}} \right] \left(\dfrac{a + 4}{a - 4}\right)$
$= \left[\dfrac{- \left(a^2 - 8a + 16\right)}{4a}\right] \left[\dfrac{a - 4}{a - 4} - \dfrac{\left(a + 4\right)^{\frac{-1}{2} + \frac{3}{2}}}{a - 4}\right] \left(\dfrac{a + 4}{a - 4}\right)$
$= \dfrac{-\left(a - 4\right)^2}{4a} \left(1 - \dfrac{a + 4}{a - 4}\right) \left(\dfrac{a + 4}{a - 4}\right)$
$= \dfrac{-\left(a - 4\right)^2}{4a} \left(\dfrac{-8}{a - 4}\right) \left(\dfrac{a + 4}{a - 4}\right)$
$= \dfrac{2 \left(a + 4\right)}{a}$