Automorphisms of unstable P¹-bundles

Let P→ X be a P¹-bundle over a variety X. The aim of this note is to understand all connected, algebraic groups Aut^∘(P)⊂ G⊂ Bir( X× P¹). We get a quite complete answer if Aut^∘(X) is a maximal, connected, algebraic subgroup of Bir(X), and P is sufficiently unstable. This gives examples of connected, algebraic subgroups of Bir( P⁴) that are not contained in any maximal one.