Structure of Connected Solvable Algebraic Groups
Let \(B\) be a connected solvable linear algebraic group over an algebraically closed field. Its unipotent radical \(R_u(B)\) is a closed connected normal unipotent subgroup, the quotient \(B/R_u(B)\) is a torus, and every maximal torus \(T\subseteq B\) maps isomorphically onto this quotient.
Moreover, maximal tori of \(B\) are conjugate by elements of \(R_u(B)\), and for any maximal torus \(T\), multiplication gives an isomorphism of varieties \[ T\times R_u(B)\xrightarrow{\sim} B. \]
Proof
By Lie-Kolchin, embed \(B\) into an upper triangular group. The subgroup acting trivially on each successive one-dimensional quotient of the standard flag is unipotent, closed, connected, and normal; it is maximal among connected normal unipotent subgroups, hence equals \(R_u(B)\).
The quotient by this subgroup acts faithfully and diagonally on the successive quotients of the flag, so it is diagonalizable. Connectedness makes it a torus. A maximal torus \(T\) maps onto \(B/R_u(B)\), and its intersection with \(R_u(B)\) is trivial, so the map is an isomorphism.
The conjugacy of maximal tori inside \(B\) follows by applying the same triangular form and eliminating off-diagonal entries by conjugation with the unipotent radical. The product map is bijective with regular inverse induced by the quotient projection and the chosen torus section, hence is an isomorphism of varieties.