Different ways to solve nested radicals with cubic roots

I want to obtain the result of:
$$\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}$$
Which turns out to be 1. Now, let's prettend we don't know what the result is. I solved it by stating
$$\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}=z$$
Then by cubing the equation:
$$4-3\biggr(\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2} \biggr)=z^3$$
$$z^3+3z-4=0$$

Now, just by an inexplicable mysticism, the equation can be restated as:
$$(z-1)(z^2+z+4)=0$$
Therefore, $z=1$, which is what I wanted to prove.

Are there another ways to solve this problem? I find this method quite impractical and not so elegant. I'm interested in ways to solve it that are MUCH simpler.

Thanks in advance!