I have the following situation (following the proof that every linear compact group is algebraic from Vinberg, Gorbatsevich and Onishchik "Lie Groups and Lie algebras III" Chapter 4 Theorem 2.1):

Let $\rho : G \rightarrow GL(V)$ be a faithful linear representation (say over the reals or complex) of $G$ such that the orbits $G\cdot v$ for $v\in V$ are algebraic varieties in $V$. Then $G$ is an algebraic subgroup of $GL(V)$.

I cannot get how to rebuild algebraic information from the orbits. If I have a morphism of $V$ I can tell for each element of a basis if its image is in its orbit but I could have all diferent elements of $G$ acting in each basis vector...


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