just a value

We know $0$ is not a solution, since $0^2 = 0.$ Therefore we can divide equation 1 by $a$ to get $a^2 = 1.$ The two solutions to this equation are $-1$ and $1$ , but $1$ does not work since $1^2 = 1.$
$-1$ does work since $(-1)^2 = 1 \neq -1.$ Therefore the answer is $a = -1$

$a^3 - a = 0\\ a(a^2 - 1) = 0\\ a = 0 (\text{or}) a^2 - 1 = 0\\ a = 0 (\text{or}) a = 1 (\text{or}) a = -1$

But if $a = 0$ and $a = 1$ , $a = a^2$ .
So, $a = \color{#69047E}{\boxed{-1}}$ .

only -1 satisfies the given condition.

a= $a^3$ => 1= $a^2$ => a = -1 or +1 and a<> $a^2$ so the answer a=-1

only -1 satisfies the given condition.