  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.
