Factorize:

$a^{3}-\frac{1}{a^{3}}-2 a+\frac{2}{a}=\left(a^{3}-\frac{1}{a^{3}}\right)-2\left(a-\frac{1}{a}\right)$

$=\left[(a)^{3}-\left(\frac{1}{a}\right)^{3}\right]-2\left(a-\frac{1}{a}\right)$

$=\left(a-\frac{1}{a}\right)\left[a^{2}+a \times \frac{1}{a}+\left(\frac{1}{a}\right)^{2}\right]-2\left(a-\frac{1}{a}\right)$

$=\left(a-\frac{1}{a}\right)\left(a^{2}+1+\frac{1}{a^{2}}\right)-2\left(a-\frac{1}{a}\right)$

$=\left(a-\frac{1}{a}\right)\left(a^{2}+1+\frac{1}{a^{2}}-2\right)$

$=\left(a-\frac{1}{a}\right)\left(a^{2}-1+\frac{1}{a^{2}}\right)$