On the exponential map of borei subgroups in a semi-simple lie group

Let G be a connected, complex, semi-simple Lie group Let g be an element in G. Let B be a Borel subgroup of G and g in B. Let m and n be the least positive integers such that the element g^m lies on a one-parameter subgroup in G and the element gⁿ lies on a one-parameter subgroup in B. We denote these integers by ind _G (g) and ind _B (g). In this note we prove the conjecture ind _G (g) = ind _B (g), if g is regular.