Question: What was an island that Guðrøðr Óláfsson ruled over?

Context paragraphs:
Olaf the Black: Óláfr Guðrøðarson (] ), commonly known in English as Olaf the Black, was a mid 13th century sea-king who ruled the Isle of Man (Mann) and parts of the Hebrides.  Óláfr was the son of Guðrøðr Óláfsson, King of the Isles, King of Dublin, and his wife Finnguala, granddaughter of Muirchertach Mac Lochlainn, High King of Ireland, King of Cenél nEógain.  Óláfr was a younger son of his father; his elder brother Rögnvaldr more than likely had a different mother.  According to the "Chronicle of Mann", Guðrøðr appointed Óláfr as heir since he had been born "in lawful wedlock".  Whether or not this is the case, on Guðrøðr's death in 1187 the Manxmen instead appointed Rögnvaldr as king, as he was a capable adult and Óláfr was a mere child.  Rögnvaldr ruled the Crovan dynasty's island-kingdom for almost 40 years, during which time the half-brothers vied for the kingship.

Rǫgnvaldr Guðrøðarson: Rǫgnvaldr Guðrøðarson (died 14 February 1229) ruled as King of the Isles from 1187 to 1226.  He was the eldest son of Guðrøðr Óláfsson, King of Dublin and the Isles.  Although the latter may have intended for his younger son, Óláfr, to succeed to the kingship, the Islesmen chose Rǫgnvaldr, who was likely Óláfr's half-brother.  Rǫgnvaldr went on to rule the Kingdom of the Isles for almost forty years before losing control to Óláfr.

Kingdom of the Isles: The Kingdom of the Isles comprised the Hebrides, the islands of the Firth of Clyde and the Isle of Man from the 9th to the 13th centuries AD.  The islands were known to the Norse as the "Suðreyjar", or "Southern Isles" as distinct from the "Norðreyjar" or Northern Isles of Orkney and Shetland.  The historical record is incomplete, and the kingdom was not a continuous entity throughout the entire period.  The islands concerned are sometimes referred to as the Kingdom of Mann and the Isles, although only some of the later rulers claimed that title.  At times the rulers were independent of external control, although for much of the period they had overlords in Norway, Ireland, England, Scotland or Orkney.  At times there also appear to have been competing claims for all or parts of the territory.  The islands involved have a total land area of over 8300 km2 and extend for more than 500 km from north to south.

Mac Scelling: Mac Scelling (fl.  1154–1173/1174), also known as Mac Scilling, was a prominent twelfth-century military commander engaged in conflicts throughout Ireland.  He is first recorded in 1154 commanding the maritime forces of Muirchertach Mac Lochlainn, King of Cenél nEógain in a bloody encounter against Toirrdelbach Ua Conchobair, King of Connacht.  Muirchertach's naval forces were drawn from the western peripheries of Scotland and the Isles.  He next appears on record in 1173/1174, supporting the cause of Ruaidrí Ua Conchobair, King of Connacht against the English colonisation of Mide.  An early modern Scottish source claims that a man of the same name was a bastard son of Somairle mac Gilla Brigte, King of the Isles.  If Mac Scelling was indeed related to Somairle, this relationship could cast light on the latter's conflict with Guðrøðr Óláfsson, King of the Isles, a man who appears to have opposed Muirchertach at some point in his career.  Although not termed so in contemporary sources, Mac Scelling may be regarded as an early archetype of later gallowglasses, heavily-armed Scottish mercenaries recruited by Irish rulers in centuries that followed.

Guðrøðr Óláfsson: Guðrøðr Óláfsson (died 10 November 1187) was a twelfth-century ruler of the kingdoms of Dublin and the Isles.  Guðrøðr was a son of Óláfr Guðrøðarson, King of the Isles and Affraic, daughter of Fergus, Lord of Galloway.  Throughout his career, Guðrøðr battled rival claimants to the throne, permanently losing about half of his realm to a rival dynasty in the process.  Although dethroned for nearly a decade, Guðrøðr clawed his way back to regain control of a partitioned kingdom, and proceeded to project power into Ireland.  Although originally opposed to the English invasion of Ireland, Guðrøðr adeptly recognised the English ascendancy in the Irish Sea region and aligned himself with the English.  All later kings of the Crovan dynasty descended from Guðrøðr.

Guðrøðr Magnússon: Guðrøðr Magnússon (fl.  1275), also known as Godred Magnusson, was an illegitimate son of Magnús Óláfsson, King of Mann and the Isles.  In 1275, whilst Mann was under Scottish overlordship, Guðrøðr led an unsuccessful revolt on the island.  According to a near contemporary source, over five hundred people lost their lives in the subsequent Scottish invasion and suppression of the Manx.  It is not certain whether Guðrøðr escaped the decisive defeat of the rebels at Battle of Ronaldsway with his life or if he was among the slain.

Niall Mac Lochlainn: Niall Mac Lochlainn (died 1176) was a king of Cenél nEógain and Cenél Conaill.  He was a member of the Meic Lochlainn, and a son of Muirchertach Mac Lochlainn, King of Cenél nEógain (died 1166).  Ruaidrí Ua Conchobair, King of Connacht (died 1198) divided Tír nEógain between Niall and Áed Méith Ua Néill (died 1230).  Muirchertach's granddaughter, Findguala, who married Guðrøðr Óláfsson, King of Dublin and the Isles (died 1187), appears to have been a daughter of Niall.

Assassination of Ingimundr: Ingimundr, also known as Ingimund, and Ingemund, was an eleventh-century delegate of Magnús Óláfsson, King of Norway.  In the last decade of the eleventh century, Ingimundr was tasked by Magnús to take control of the Kingdom of the Isles.  The realm had descended into utter chaos after the death of Guðrøðr Crovan, King of the Isles in 1095, which was followed by kin-strife amongst Guðrøðr's descendants, and the encroachment of Irish authority into the region.  Ingimundr and his followers were slain in Lewis by the leading Islesmen whilst he was in the midst of securing the kingship.  The following year, Magnús took matters into his own hands, and personally oversaw the conquest of the Isles himself.

Guðrøðr Rǫgnvaldsson: Guðrøðr Rǫgnvaldsson (died 1231), also known as Guðrøðr Dond, was a thirteenth-century ruler of the Kingdom of the Isles.  He was a member of the Crovan dynasty, and a son of Rǫgnvaldr Guðrøðarson, King of the Isles, the eldest son of Guðrøðr Óláfsson, King of Dublin and the Isles.  Although the latter may have intended for his younger son, Óláfr, to succeed to the kingship, the Islesmen instead settled upon Rǫgnvaldr, who went on to rule the Kingdom of the Isles for almost forty years.  The bitterly disputed royal succession divided the Crovan dynasty for three generations, and played a central role in Guðrøðr's recorded life.

Lǫgmaðr Guðrøðarson: Lǫgmaðr Guðrøðarson (] ) was a late eleventh-century King of the Isles, whose rise, reign, and fall from power are obscure.  He was the eldest son of Guðrøðr Crovan, King of Dublin and the Isles, a Norse-Gaelic dynast who conquered and ruled the kingdoms of the Isles and Dublin, before dying in 1095.  Three years after the latter's death, the Isles was conquered by Magnús Óláfsson, King of Norway, whose regime in the region lasted until his death in 1103.  The chronology of Lǫgmaðr's reign is uncertain: he may have begun his reign either before Magnús' conquest, during his regime, or after his demise.

Supporting paragraphs: Guðrøðr Óláfsson Kingdom of the Isles
Answer: the Isle of Man