# Rebuild a linear algebraic group from its orbits

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...