Let $K/\mathbb{Q}$ be a degree $4$ number field. Is it known how to determine whether the Galois closure of $K$, say $K'$, contains the 4-th roots of unity?

In the cubic case, there is a succinct answer: the Galois closure $E'$ of a cubic field $E/\mathbb{Q}$ contains the third roots of unity if and only if $E$ is a pure cubic field: that is, $E = \mathbb{Q}(\sqrt[3]{n})$ for some cube-free integer $n$.

Such a simple characterization doesn't seem to work for the quartic case, since for example the field $\mathbb{Q}(\sqrt{a}, \sqrt{-1})$ with a square-free integer $a \ne \pm 1$ contains the $4$-th roots of unity but is not of the form $\mathbb{Q}(\sqrt[4]{n})$ for a $4$-free integer $n$.

If this characterization is known classically, can someone give a reference?