It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L of V such that That is, ρ(G) has an invariant line L, on which G therefore acts through a one-dimensional representation.
This is equivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for all
It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one.
is conjugate in GL(n,K) (where n = dim V) to a subgroup of the group T of upper triangular matrices, the standard Borel subgroup of GL(n,K): the image is simultaneously triangularizable.
The theorem applies in particular to a Borel subgroup of a semisimple linear algebraic group G. If the field K is not algebraically closed, the theorem can fail.
The standard unit circle, viewed as the set of complex numbers
of absolute value one is a one-dimensional commutative (and therefore solvable) linear algebraic group over the real numbers which has a two-dimensional representation into the special orthogonal group SO(2) without an invariant (real) line.