On the descent of Levi factors. (Q1936513)

Let \(G\) be a linear algebraic group over a field \(k\) of characteristic \(p>0\), and suppose that the unipotent radical \(R\) of \(G\) is defined and split over \(k\). By a Levi factor of \(G\), one means a closed subgroup \(M\) which is a complement to \(R\) in \(G\). Two results related to the descent of Levi factors are given. First, suppose \(\ell\) is a finite Galois extension of \(k\) for which the extension degree \([\ell:k]\) is relatively prime to \(p\). If \(G_{/\ell}\) has a Levi decomposition, then \(G\) has a Levi decomposition. Second, suppose that there is a \(G\)-equivariant isomorphism of algebraic groups \(R\simeq\text{Lie}(R)\) -- i.e. \(R\) is a vector group with a linear action of the reductive quotient \(G/R\). If \(G_{/k_{\text{sep}}}\) has a Levi decomposition for a separable closure \(k_{\text{sep}}\) of \(k\), then \(G\) has a Levi decomposition. Finally, an example of a disconnected, Abelian, linear algebraic group \(G\) for which \(G_{/k_{\text{sep}}}\) has a Levi decomposition, but \(G\) itself has no Levi decomposition, is given.