Quadratic forms invariant under algebraic groups

For representations of semisimple Lie algebras over the complex numbers, there is a well-known criterion in terms of highest weights for whether the representation has a quadratic form that is invariant under the action of the Lie algebra, i.e., whether the representation is orthogonal. If it is, then the invariant quadratic form is
uniquely determined up to a scalar. These results are easily extended to irreducible representations of semisimple algebraic groups over arbitrary fields. This is true also for non-split groups, and the quadratic form gives some information about the group. A famous example is the Killing form for simple Lie algebras over the real
numbers. It turns out that these invariant quadratic forms are already interesting for groups of type A1.