How to construct cube roots with an angle trisector

Question

Is there a construction that, given lenght $l$, allows you to construct $\sqrt[3] l$? I think I have read that the trisector only allows for constructing roots of cubic equations with 3 real roots, whereof $x^3-l=0$ is not a example.
What tools do you need then to construct $\sqrt[3] l$ for arbitrary $l$?