Length functions exponentially distorted on subgroups of complex Lie groups

We introduce a notion of a length function exponentially distorted on a (compactly generated) subgroup of a locally compact group. We prove that for a connected linear complex Lie group there is a maximum equivalence class of length functions exponentially distorted on a normal integral subgroup lying between the exponential and nilpotent radicals. Moreover, a function in this class admits an asymptotic decomposition similar to that previously found by the author for word length functions, i.e., in the case of exponential radical [J. Lie Theory 29:4, 1045--1070, 2019]. In the general case we use auxiliary length functions constructed via holomorphic homomorphisms to Banach PI-algebras.