Tul of Moray - Sources?

What are the Sources for Tul?

I can find this (below):http://www.clanmacfarlanegenealogy.info/genealogy/TNGWebsite/getper..., which cites:
* Kings & Queens, Neil Grant, (pub 2003 by HarperCollinsPublishers Hammersmith London W6 8JB), p10 (Reliability: 3)
* Pedigrees of the Scottish Clans, John D McLaughlin, (http://members.aol.com/lochlan4/pedigree.htm), Genealogy of the Clan Duff (Reliability: 3)
*Kings & Queens, Neil Grant, (pub 2003 by HarperCollinsPublishers Hammersmith London W6 8JB), p11 (Reliability: 3)

=King Lulach married Fionghuala, daughter of Sinel, Mormaor of Angus. They had two children , Mael-Snechtai, afterwards, provincial King of Moray, and Princess Olith (some say "Tul"). The details of Lulach's enthronement and coronation at Scone Abbey in August 1057 are the first actually recorded in Scottish and Irish annals. He is believed to be one of the first Scottish kings who was actually crowned at his enthronement- since enthronement on the Stone of Destiny rather than crowning was the legal act of king-making in Scotland. Crowning was a later development.

Mael-Snechtai laid claim to the Scottish Throne in 1078, and the Anglo-Saxon chronicles report that King Mael-Coluim III raided Mael-Schectai's compound , seized his mother, Queen Dowager Fionghuala, and all his possessions. Mael-Schechtai himself, barely escaped. Now ruined, he entered a monastery where he died unmarried in 1085. Some writers suggest Mael-Schechtai may have been forced into a monastery by Mael-Coluim III. Mael-Schechtai's sister, Olith, married, according to Mackay tradition, her own cousin, Aedh, Mormaor of Ross, who according to Michael Mackay, was a male-line cousin of the late King Lulach descended from King Lulach's grand-uncle, Prince Domnall mac Ruaridh of Moray. Aedh succeeded Mael-Snechtai as Mormaor (not king) of Moray. He appears to have lived at peace with King Mael-Coluim's family, since he witnessed several royal charters. Aedh and Princess Olith had three sons. These sons revived their uncle's claim to the Scottish Throne. The eldest of these, Oengus, took the title of King of Moray on his father's death, and led an army south into Scotland proper in 1130. There he was slain -some say in single combat with a Norman knight- at the Battle of Inchbare. His brother MaelColuim , Earl of Ross, fought and was imprisoned by King Mael-Coluim IV. This king also expelled the Moray royal family from Moray, along with most of Morays' native inhabitants in a campaign which was conducted over about five years.

The family resettled in Strathnaver, Sutherland where they founded a clan named Clan Aedh (sometimes referred to as Clan Morggan- Lulach's royal ancestor). The present Chief of Clan Aedh, Lord Hugh Mackay of Reay, is the legal representative of King Lulach's family, although his DNA has not yet been tested. The Chiefs of Clan Gregor (MacGregor) and Finguine (Mackinnon) are descended in the male line from Bishop Cormac of Dunkeld, who, according to the Irish Annals of Tigernach, was a great/great-grandson of King MacBeatha=