Algebraic one-parameter subgroup ($\lambda :{\mathbb{G}}_m\to G$)

An algebraic one-parameter subgroup is a regular group homomorphism from the multiplicative algebraic group. In $S{L}_n$, a change of basis makes it $t\mapsto \mathrm{diag}(t^{a_1},\dots ,t^{a_n})$ for integers with sum zero. This is the algebraic multiplicative notion, distinct from an additive real-parameter subgroup of a Lie group.