Eta/Chronology

Background
At some point, Eta was afflicted by Demon Possession, which eventually led to her crossing paths with Shadow and being recruited into Shadow Garden by him.

First Assault on the Cult
After her master's elder sister, Claire Kagenou, was abducted by Viscount Grease serving the Cult of Diablos, Eta participated in the assault on the secret base where Claire was being held. Advancing alongside Shadow and the other Seven Shadows, they managed to rescue Claire after eliminating all Cult members and branch leader Grease. Afterward, Eta would join her sisters in leaving Shadow for two years to expand their influence to keep pace with the Cult's power.

Ancient Capital City Reconstruction
After the process of relocating to Alexandria, Eta was ordered by Alpha to build a defense system together with Lambda.

Renowned Architect: Eta Lloyd Wright
At some point perhaps after completing the headquarters set up for Shadow Garden during the refurbishing process of Alexandria, Eta crafted her public alias of 'Eta Lloyd Wright' from the names of inventors mentioned from "Shadow Wisdom." Eta would utilize her mastery of mathematics and other sciences from Shadow's tutelage to become a renowned architect, developing materials via Mitsugoshi Company while assisting Gamma in building Mitsugoshi's main store after acquiring shares from the former company director of Luna, Ltd. in the center of the capital and implementing them into her building designs. Earning awards and accolades, Eta gained access to the nobility of the kingdoms and garnered prestige to visit ancient sites for restoration or improvement-- the latter benefit helped Shadow Garden to gain access via Mitsugoshi to enter otherwise unassailable locales.

Bushin Festival Tournament
In the anime, during the beginning of the Bushin Festival, Eta helps her lord Shadow, along with Gamma, Nu, Chi and Omega, to disguise him using a machine that molds the head of a person into someone else. With Shadow in the disguise he wanted, she retired to resume sleeping, wheeled off on a box dolly.

By the end of the festival, Eta communicated with the other Shadow Garden members in Oriana Kingdom using Morse code. Establishing a coded covert communications network across the continent.

Red Moon Incident
Eta was theorizing about the similarity between the Demon Possession and the Vampire Progenitor's blood, and informed Beta to look for information in the Crimson Tower of the Lawless City. Afterward, they started preparing an operation because their master intended to go there.

After the Blood Queen's rampage ended, Beta was able to recover the sample of Progenitor's blood and had already collected information from the tower's library to study further of her own theory's research of Demon Possession.

The research was proven successful as Alpha was able to use the 'Red Mist Form' technique. Able to use it against John Smith later, and Zeta using 'Black Mist Form' during her fight with Delta.

Great Trading Company Conflict
Following the events of Delta's disappearance, Eta was given an encrypted message by Beta from Shadow written in an alien language (actually Japanese). After Eta decoded the writing, it was discovered that the message was Shadow's explanation on his "betrayal" and the deciphered parcel was forwarded to Alpha and the other Seven Shadows.

Return to Earth
After Beta returned from her adventure in another world with Shadow, she brought an alien inhabitant from that world back with her. Alpha subsequently placed the Otherworlder under Beta's protective custody out of fear of what Eta would do if she got her hands on the alien.