Inspector reported procedural errors caused a thrust reverser sleeve to be installed without correct certifications and tracking information after being removed serviceable (robbed) for installation on another aircraft.

Narrative

Day 0; maintenance management robbed #1 engine LH reverser translating sleeve from a serviceable #1 engine LH thrust reverser assembly Company part number (P/N) XX-XXXX-X-XXXX; this thrust reverser was made serviceable over a month prior on Repair Order# XXXXXXX.Robbed #1 engine LH thrust reverser sleeve Company P/N XX-XXXX-X-XXXY; was installed on Aircraft X without maintenance management following General Maintenance Manual (GMM) XX-XX-XX process; that is to fill out robbed unit tag and to make a serviceable tag after verifying the serviceable condition of the removed part. Also per GMM; the quality department Inspector is required to complete computer transactions to update removal and installation record of rotable part.Stores receiving storekeeper was notified to make serviceable tag for robbed #1 engine LH thrust reverser translating sleeve after Technician installed on aircraft (storekeeper is not responsible for the serviceability of any robbed part; that is the responsibility of the Technician removing the robbed part).Storekeeper also completed the removal transaction in the computer system for the robbed #1 engine LH reverser sleeve.Maintenance management; stores never followed the GMM XX-XX-XX process and quality department failure to support the removal and installation process (computer transactions; off log and on log of thrust reverse sleeve).GMM XX-XX-XX Item-XA Pg.1 through Pg.10 basically states the Technician removing or installing a robbed part or component from an aircraft is responsible for compliance as outline in this GMM; also GMM Item XA (13) states robbing a rotable component from another rotable component such as engine; APU or landing gear; requires computer transactions. Signee assigned to the component installation will be responsible to complete all required entries (This was not accomplished by quality department).Making leadership responsible for not adhering to GMM procedures; too many departments never read the GMM on robbed components; putting pressure on workers just to get the aircraft back in service regardless of the situation; even when they have no knowledge of what the process is. For example the storekeepers who have never got involved in robbed parts before and are now told it is now their job. The Company needs to stop taking short cuts and get serious with keeping their standards at a high level.