 Normal bundle of a smooth plane conic

Embed a smooth plane conic in ${\mathbf{P}}^3$ through a plane. The plane equation and conic equation form a regular sequence of degrees one and two, so its normal sheaf is ${\mathcal{O}}_C(1)\oplus {\mathcal{O}}_C(2)$. Under $C\cong {\mathbf{P}}^1$, hyperplane degree is two, hence this normal bundle is ${\mathcal{O}}_{{\mathbf{P}}^1}(2)\oplus {\mathcal{O}}_{{\mathbf{P}}^1}(4)$. The normal exact sequence splits as well because $H^1({\mathbf{P}}^1,\mathcal{O}(2))=0$.