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The LUCIFER project aims at deploying the first array of enrichedscintillating bolometers for the investigation of neutrinoless double-betadecay of $^{82}$Se. The matrix which embeds the source is an array of ZnSecrystals, where enriched $^{82}$Se is used as decay isotope. The radiopurity ofthe initial components employed for manufacturing crystals, that can beoperated as bolometers, is crucial for achieving a null background level in theregion of interest for double-beta decay investigations. In this work, weevaluated the radioactive content in 2.5 kg of 96.3\% enriched $^{82}$Se metal,measured with a high-purity germanium detector at the Gran Sasso deepunderground laboratory. The limits on internal contaminations of primordialdecay chain elements of $^{232}$Th, $^{238}$U and $^{235}$U are respectively:$<$61 $\mu$Bq/kg, $< $110 $\mu$Bq/kg and $<$74 $\mu$Bq/kg at 90\% C.L.. Theextremely low-background conditions in which the measurement was carried outand the high radiopurity of the $^{82}$Se allowed us to establish the moststringent lower limits on the half-lives of double-beta decay of $^{82}$Se to0$^+_1$, 2$^+_2$ and 2$^+_1$ excited states of $^{82}$Kr of 3.4$\cdot$10$^{22}$y, 1.3$\cdot$10$^{22}$ y and 1.0$\cdot$10$^{22}$ y, respectively, with a 90\%C.L..

Double-beta decay investigation with highly pure enriched $^{82}$Se for the LUCIFER experiment