Johnson, you have one of the foremost experts in the world on such matters (over general fields) just upstairs from your office. Make use of that.

Many of the basic constructions work for split groups over fields, but proving good properties (such as irreducibility and classification results) requires being over a field of characteristic 0. (Once constructions are made, to prove things one can extend scalars to an algebraic closure, or even reduce to the familiar case over $\mathbf{C}$ if so inclined, by the "Lefschetz Principle".) The Lie algebra is a good invariant (e.g., faithful!) over fields of characteristic 0, but even then it only retains at best information about groups up to isogeny. Another case where it gives a useful invariant is over $\mathbf{Z}/p\mathbf{Z}$-algebras where, together with the $p$-Lie algebra structure, it gives an equivalence with the category of finite locally free groups with vanishing relative Frobenius morphism (loosely speaking because such vanishing allows one to get by with truncated exponential in degrees $< p$); this is explained in SGA3, VII$_{\rm{A}}$, 7.2, 7.4

More generally one cannot expect to get mileage out of the Lie algebra alone (but it is still perfectly useful via its role in classification by means of root systems, among other things).

Johnson, you have one of the foremost experts in the world on such matters (over general fields) just upstairs from your office. Make use of that.

Many of the basic constructions work for split groups over fields, but proving good properties (such as irreducibility and classification results) requires being over a field of characteristic 0. (Once constructions are made, to prove things one can extend scalars to an algebraic closure, or even reduce to the familiar case over $\mathbf{C}$ if so inclined, by the "Lefschetz Principle".) The Lie algebra is a good invariant (e.g., faithful!) over fields of characteristic 0, but even then it only retains at best information about groups up to isogeny. Another case where it gives a useful invariant is over $\mathbf{Z}/p\mathbf{Z}$-algebras where, together with the $p$-Lie algebra structure, it gives an equivalence with the category of finite locally free groups $G$ with vanishing relative Frobenius morphism such that the sheaf of invariant 1-forms is locally free over the base (loosely speaking because such vanishing allows one to get by with truncated exponential in degrees $< p$); this is explained in SGA3, VII$_{\rm{A}}$, 7.2, 7.4

More generally one cannot expect to get mileage out of the Lie algebra alone (but it is still perfectly useful via its role in classification by means of root systems, among other things).

Johnson, you have one of the foremost experts in the world on such matters (over general fields) just upstairs from your office. Make use of that.

Many of the basic constructions work for split groups over fields, but proving good properties (such as irreducibility and classification results) requires being over a field of characteristic 0. (Once constructions are made, to prove things one can extend scalars to an algebraic closure, or even reduce to the familiar case over $\mathbf{C}$ if so inclined, by the "Lefschetz Principle".) The Lie algebra is only a good invariant (e.g., faithful!) over fields of characteristic 0, and even then it only retains at best information about groups up to isogeny. The one other case where it gives a useful invariant is over $\mathbf{Z}/p\mathbf{Z}$-algebras where, together with the $p$-Lie algebra structure, it gives an equivalence with the category of finite locally free groups with vanishing relative Frobenius morphism (loosely speaking because such vanishing allows one to get by with truncated exponential in degrees $< p$); this is explained in SGA3, VII$_{\rm{A}}$, 7.2, 7.4

Johnson, you have one of the foremost experts in the world on such matters (over general fields) just upstairs from your office. Make use of that.

Many of the basic constructions work for split groups over fields, but proving good properties (such as irreducibility and classification results) requires being over a field of characteristic 0. (Once constructions are made, to prove things one can extend scalars to an algebraic closure, or even reduce to the familiar case over $\mathbf{C}$ if so inclined, by the "Lefschetz Principle".) The Lie algebra is a good invariant (e.g., faithful!) over fields of characteristic 0, but even then it only retains at best information about groups up to isogeny. Another case where it gives a useful invariant is over $\mathbf{Z}/p\mathbf{Z}$-algebras where, together with the $p$-Lie algebra structure, it gives an equivalence with the category of finite locally free groups with vanishing relative Frobenius morphism (loosely speaking because such vanishing allows one to get by with truncated exponential in degrees $< p$); this is explained in SGA3, VII$_{\rm{A}}$, 7.2, 7.4

More generally one cannot expect to get mileage out of the Lie algebra alone (but it is still perfectly useful via its role in classification by means of root systems, among other things).