We work over $k=\mathbb{C}$. We consider the the Grassmanian $G(2,4)$ of lines $\mathbb P^3$ which we embed by Plucker into $\mathbb P^5$. It is basic that under this embedding $G(2,4)$ is isomorphic with a $4$-dimnl quadric $Q^4 \subset \mathbb P^5$.

Now, take any curve $C'$ in $\mathbb P^3$ and associate to it a curve $C$ in $G(2,4)$ consisting of the collection of lines tangent to $C'$. In this answer Dmitri Panov claims that after this Plucker identification of $G(2,4)$ with $Q^4$ it can be shown that the tangent variety $TC$ of $C$ is contained in $Q^4$.

Question: How to see it? Is it possible to prove it purely algebraically? Indeed as sketched in the comments below the linked answer it's possible to show it analytically working with local parametrization of the curve $C'$. But I'm wondering isn't it possible to give a pure algebraic proof of it? At all, that seems not to be hard, but I can't fiddle it out.

We work over $k=\mathbb{C}$. We consider the the Grassmanian $G(2,4)$ of lines in $\mathbb P^3$ which we embed by Plücker into $\mathbb P^5$. It is basic that under this embedding $G(2,4)$ is isomorphic with a $4$-dimnl quadric $Q^4 \subset \mathbb P^5$.

Now, take any curve $C'$ in $\mathbb P^3$ and associate to it a curve $C$ in $G(2,4)$ consisting of the collection of lines tangent to $C'$. In this answer to quadrics containing the tangential variety of a curve Dmitri Panov claims that after this Plücker identification of $G(2,4)$ with $Q^4$ it can be shown that the tangent variety $TC$ of $C$ is contained in $Q^4$.

Question: How to see it? Is it possible to prove it purely algebraically? Indeed as sketched in the comments below the linked answer it's possible to show it analytically working with local parametrization of the curve $C'$. But I'm wondering isn't it possible to give a pure algebraic proof of it? At all, that seems not to be hard, but I can't fiddle it out.