A similar construction works using a primitive nontrivial purely inseparable finite extension of any imperfect field in any positive characteristic, the only difference being that the formula for the norm map is a bit more complicated than in the preceding quadratic examples.
More generally, if K is a non-trivial purely inseparable finite extension of k and G is any non-trivial connected reductive K-group defined then the Weil restriction H=RK/k(G) is a smooth connected affine k-group for which there is a (surjective) homomorphism from HK onto G. The kernel of this K-homomorphism descends the unipotent radical of the geometric fiber of H and is not defined over k (i.e., does not arise from a closed subgroup scheme of H), so RK/k(G) is pseudo-reductive but not reductive.
The commutative pseudo-reductive groups admit no useful classification (in contrast with the connected reductive case, for which they are tori and hence are accessible via Galois lattices), but modulo this one has a useful description of the situation away from characteristics 2 and 3 in terms of reductive groups over some finite (possibly inseparable) extensions of the ground field.
Moreover, in characteristic 2 there are additional possibilities arising not from exceptional isogenies but rather from the fact that for simply connected type C (I.e., symplectic groups) there are roots that are divisible (by 2) in the weight lattice; this gives rise to examples whose root system (over a separable closure of the ground field) is non-reduced; such examples exist with a split maximal torus and an irreducible non-reduced root system of any positive rank over every imperfect field of characteristic 2.
Subsequent work of Conrad & Prasad (2016), building on additional material included in the second edition Conrad, Gabber & Prasad (2015), completes the classification in characteristic 2 up to a controlled central extension by providing an exhaustive array of additional constructions that only exist when [k:k^2]>2 , ultimately resting on a notion of special orthogonal group attached to regular but degenerate and not fully defective quadratic spaces in characteristic 2.