Weyl group of a maximal torus in a linear algebraic group

Definition
Suppose $$G$$ is a linear algebraic group over a field $$K$$ and $$T$$ is a maximal torus in $$G$$. The Weyl group of $$T$$, denoted $$W(T)$$, is defined as follows:


 * As an abstract group, it is the defining ingredient::Weyl group of $$T$$ in $$G$$, which can explicitly be defined as the quotient group of the normalizer $$N_G(T)$$ by the centralizer $$C_G(T)$$.
 * As a subgroup of $$G$$, it can be defined as any permutable complement to $$C_G(T)$$ in $$N_G(T)$$. (Do such complements always exist?). Any such subgroup is termed a Weyl subgroup.

Note that if $$K$$ is an algebraically closed field, then $$T$$ is unique up to conjugacy in $$G$$, hence $$W(T)$$ is uniquely determined up to isomorphism by $$G$$.

Particular cases
We denote the underlying field by $$K$$.