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Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...

$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...

The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...

I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras. Let $G$ be a reductive group ...