Question: Yelü Chucai was a retainer of which founder of the Mongol Empire?

Context articles: Berke–Hulagu war
Mongol invasions of Georgia
Genghis Khan
House of Ögedei
Mongol conquest of China
Yelü Chucai
Manchuria under Yuan rule
List of Khitan inscriptions
Ariq Böke
Timeline of the Mongol Empire

The Berke–Hulagu war was fought between two Mongol leaders, Berke Khan of the Golden Horde and Hulagu Khan of the Ilkhanate.
 It was fought mostly in the Caucasus mountains area in the 1260s after the destruction of Baghdad in 1258.
 The war overlaps with the Toluid Civil War in the Mongol Empire between two members of the Tolui family line, Kublai Khan and Ariq Böke, who both claimed the title of Great Khan (Khagan).
 Kublai allied with Hulagu, while Ariq Böke sided with Berke.
 Hulagu headed to Mongolia for the election of a new Khagan to succeed Möngke Khan, but the loss of the Battle of Ain Jalut to the Mamluks forced him to withdraw back to the Middle East.
 The Mamluk victory emboldened Berke to invade the Ilkhanate.
 The Berke–Hulagu war and the Toluid Civil War as well as the subsequent Kaidu–Kublai war marked a key moment in the fragmentation of the Mongol empire after the death of Möngke, the fourth Great Khan of the Mongol Empire.
Mongol conquests of Kingdom of Georgia, which at that time consisted of Georgia proper, Armenia, and much of the Caucasus, involved multiple invasions and large-scale raids throughout the 13th century.
 The Mongol Empire first appeared in the Caucasus in 1220 as generals Subutai and Jebe pursued Muhammad II of Khwarezm during the destruction of the Khwarezmian Empire.
 After a series of raids in which they defeated the Georgian and Armenian armies (Battle of Caucasus Mountain), Subutai and Jebe continued north to invade Kievan Rus'.
 After his empire was destroyed, Khwarazm ruler Jalal ad-Din Mingburnu, son of Muhammed II, battled both the Mongols and the Georgians before moving on to challenge the Seljuks in Anatolia.
 A full-scale Mongol conquest of the Caucasus and eastern Anatolia began in 1236, in which the Kingdom of Georgia, the Sultanate of Rum, and the Empire of Trebizond were subjugated, the Armenian Kingdom of Cilicia and other Crusader states voluntarily accepted Mongol vassalage, and the Assassins were eliminated.
 The Mongols also invaded Dzurdzuketia, modern-day Chechnya, but faced continual resistance in that area.
 After the death of Möngke Khan in 1259, the Mongol Empire descended into civil war and Berke of the Golden Horde and Hulagu of the Ilkhanate repeatedly invaded each other in the Caucasus until the ascension of Kublai Khan in 1264.
Genghis Khan (c. 1162 August 18, 1227), born Temüjin, was the Great Khan and founder of the Mongol Empire, which became the largest contiguous empire in history after his death.
 He came to power by uniting many of the nomadic tribes of Northeast Asia.
 After founding the Empire and being proclaimed "Genghis Khan", he launched the Mongol invasions that conquered most of Eurasia.
 Campaigns initiated in his lifetime include those against the Qara Khitai, Caucasus, and Khwarazmian, Western Xia and Jin dynasties.
 These campaigns were often accompanied by large-scale massacres of the civilian populations – especially in the Khwarazmian and Western Xia controlled lands.
 By the end of his life, the Mongol Empire occupied a substantial portion of Central Asia and China.
The House of Ögedei, sometimes called the Ögedeids, were an influential family of Mongol Borjigin (Imperial, or Golden Family) from the 12th to 14th centuries.
 They were descended from Ögedei Khan (1186-1241), a son of Genghis Khan who had become his father's successor, second Khagan of the Mongol Empire.
 Ögedei continued the expansion of the Mongol Empire.
 When, after the Toluid Möngke Khan's death, the Mongol Empire disintegrated into civil war, the members of the House of Ogedei were influential players in the politics of the region.
 Of Genghis Khan's sons — Ogedei, Jochi, Chagatai, and Tolui — the House of Ögedei tended to ally with the Chagataids (descendants of Chagatai) against the House of Jochi, while seeking control for themselves within the Chagatai Khanate at first.
 The Ogedeids also allied with the Golden Horde against the Yuan emperor Kublai Khan (son of Tolui), who was allied with his brother Hulagu, leader of the Ilkhanate in Persia.
 The Ogedeids attempted to unite the Mongol Empire under their own rule, and Ogedeid princes continued to march against the Yuan dynasty well into the 14th century such as during the Kaidu–Kublai war.
The Mongol conquest of China was a series of major military efforts by the Mongol Empire to invade China proper.
 It spanned six decades in the 13th century and involved the defeat of the Jin dynasty, Western Xia, the Dali Kingdom and the Southern Song.
 The Mongol Empire under Genghis Khan started the conquest with small-scale raids into Western Xia in 1205 and 1207.
 By 1279, the Mongol leader Kublai Khan had established the Yuan dynasty in China and crushed the last Song resistance, which marked the onset of all of China under the Mongol Yuan rule.
 This was the first time in history that the whole of China was conquered and subsequently ruled by a foreign or non-native ruler.
Yelü Chucai (Yeh-lu Chu-tsai; ; Mongolian: Urtu Saqal, 吾圖撒合里, "long beard"; the components of his name also variously spelt Yeh-Lu, Ye Liu, Yeliu, Chutsai, Ch'u-Ts'ai, etc.) (July 24, 1190 - June 20, 1244) was a statesman of Khitan ethnicity with royal family lineage to the Liao Dynasty, who became a vigorous adviser and administrator of the early Mongol Empire in the Confucian tradition.
 He was the first of Genghis Khan's retainers to formulate policy during the Mongol conquests, and he also introduced many administrative reforms in North China during the reign of Genghis Khan and his successor Ögedei.
Manchuria under Yuan rule refers to the Yuan dynasty's rule over Manchuria, including modern Northeast China and Outer Manchuria from the beginning to the end of the dynasty.
 Mongol rule over Manchuria was established during the Mongol Empire's conquest of the Jurchen Jin dynasty in the early 13th century.
 It became a part of the Yuan dynasty, division of the Mongol Empire, when the dynasty was founded by the Mongol leader Kublai Khan.
 Even after the overthrown of the Mongol Yuan dynasty by the Ming dynasty founded by native Chinese in 1368, Manchuria was still controlled by the Northern Yuan dynasty based in Mongolia for almost 20 years, until it was conquered by the Ming during the Ming military campaign against Naghachu and put under Ming rule.
The list of Khitan inscriptions comprises a list of the corpus of known inscriptions written in the Khitan large script and the Khitan small script.
 These two scripts were used by the Khitan people in northern China during the 10th through 12th centuries for writing the extinct Khitan language.
 The Khitan language was in use during the Liao Dynasty (907–1125), the Kara-Khitan Khanate (1124–1218) and the Jin dynasty (1115–1234), but the last recorded Khitan speaker, Yelü Chucai, died in 1243, and the language probably became extinct soon afterwards.
Ariq Böke (after 1219–1266), the components of his name also spelled Arigh, Arik and Bukha, Buka (Mongolian: Аригбөх ; Chinese: 阿里不哥), was the seventh and youngest son of Tolui, a grandson of Genghis Khan.
 After the death of his brother the Great Khan Möngke, Ariq Böke claimed the title of the Great Khan of the Mongol Empire and briefly took power while his brothers Kublai (more commonly known as Kublai Khan) and Hulagu (more commonly known as Hulagu Khan) were absent from the Mongolian homeland.
 When Kublai returned for an election in 1260, rival factions could not agree, and elected both claimants, Kublai and Ariq Böke, to the throne, resulting in the Toluid Civil War that fragmented the Mongol Empire.
 Ariq Böke was supported by the traditionalists of the Mongol Empire, while his brother Kublai was supported by the senior princes of North China and Manchuria.
This is the timeline of the Mongol Empire from 1206, when Temüjin received the title of Genghis Khan, to the death in 1227 of the last successors to Genghis as overlord of all the Mongol dominions, though after the death of Kublai Khan in 1294, their suzerainty over the other divisions of the Mongol Empire (initially the Chagatai Khanate, the Golden Horde, and the Ilkhanate) was almost notional.
 The Mongol Empire is usually considered to have come to an end in 1368, though the title of Khagan continued to be used by the rulers of the Northern Yuan dynasty, a far less powerful successor entity, until 1634.