How does $(2, 1 + \sqrt{-5})$ become principal in the Hilbert class field?

Question

Let $K = \mathbb{Q}(\sqrt{-5})$ and $L = K(\sqrt{-1})$ its Hilbert class field. Then according to the principal ideal theorem, the non-principal ideal $(2, 1+\sqrt{-5})$ becomes principal in $\mathcal{O}_L$. In the link above, it is suggested that $(2, 1 + \sqrt{-5})\mathcal{O}_L = (1 + \sqrt{-1})\mathcal{O}_L$. I can see it is reasonable, as both sides square to the ideal $(2)\mathcal{O}_L$. However, I am unable to see why $1 + \sqrt{-5}\in (1 + \sqrt{-1}) \mathcal{O}_L$.