About Máel Snechtai mac Lulaich
From the English Wikipedia page on Máel Snechtai of Moray:
Máel Snechtai of Moray (or Máel Snechtai mac Lulaich) was the ruler of Moray, and, as his name suggests, the son of Lulach, King of Scotland.
He is called on his death notice in the Annals of Ulster, "Máel Snechtai m. Lulaigh ri Muireb" (="Máel Snechtai, Lulach's son, King of Moray"), which is a significant terminological development, since previously the titles for the ruler of Moray had either been "King of Scotland" or "Mormaer." The title is repeated for his successor, Óengus (if indeed the latter were his successor).
Perhaps then the events of Máel Snechtai's caused some kind of identity disassociation between the Men of Moray and the Men of Scotland. The Anglo-Saxon Chronicle, s.a. 1078, reports that "In this year Máel Coluim seized the mother of Máel Snechtai ...and all his treasures, and his cattle; and he himself escaped [only] with difficulty."
Sadly, the ... represents a lacuna in the text one line long;  but it is clear that Máel Snechtai was defeated by King Máel Coluim III in some kind of conflict. Grant's suggestion  that Máel Snechtai subsequently retired to a monastery is based only on textual innuendo, and is made unlikely by the fact that he was called "King of Moray" on his death.
Máel Snechtai, as the son of King Lulach, undoubtedly perceived himself as the rightful king of Scotland, and the already noted conflict with Máel Coluim III strongly suggests that he pursued his claim.
His death date is based on the year given by the Annals of Ulster, s.a. 1085.
- . A.O. Anderson, Scottish Annals, p. 100.
- . A. Grant, "The Province of Ross", p. 103.
Anderson, Alan Orr, Early Sources of Scottish History: AD 500-1286, 2 Vols, (Edinburgh, 1922)
Anderson, Alan Orr, Scottish Annals from English Chroniclers: AD 500-1286, (London, 1908), republished, Marjorie Anderson (ed.) (Stamford, 1991)
Grant, Alexander, "The Province of Ross and the Kingdom of Alba", in E.J. Cowan and R.Andrew McDonald (eds.), Alba: Celtic Scotland in the Medieval Era, (Edinburgh, 2000)
GILLACOMGAIN (-burned alive 1032). Mormaer of Moray. The Annals of Ulster record that "Gilla Comgán son of Mael Brigte, earl of Moray was burned together with fifty people" in 1032. (Cawley's Medlands)
m [as her first husband, GRUOCH, daughter of BOITE--- (-)]. Her possible first marriage appears to be based on the following logic. The Continuation of the Synchronisms of Flann Mainistreach records Lulach as son of Macbeth. The Annals of Ulster record that "Lulach son of Gilla Comgain, over-king of Scotland was killed in battle by Mael Coluim son of Donnchad" in 1058. Dunbar, basing his argument on this and the other sources which are quoted in this section, states that "from the above it seems most probable that Lulach was son of Gillacomgan and step-son of Macbeth". In addition, the 12th century Cronica Regum Scottorum lists "Lulac nepos filii Boide" ["nephew of the son of Boite"] as successor of King Macbeth. However, there does not appear to be a surviving source which more specifically confirms that Macbeth´s queen was the widow of Gillacomgain and mother of Lulach. She married [secondly] ([after 1032]) Macbeth, who succeeded in 1040 as MACBETH King of Scotland. "Machbet filius Finlach…et Gruoch filia Bodhe, rex et regina Scottorum" made grants to the church of St Serf, although the document also names "Malcolmus Rex filius Duncani" which casts doubt on its authenticity. (Cawley's Medlands)
Mormaer Gillacomgain & his [wife] had [one child]: (Cawley's Medlands)
i) LULACH (-killed in battle Essie, Strathbogie 17 Mar 1058, bur Isle of Iona). As noted above, the primary sources are unclear concerning the parentage of Lulach. He succeeded [his stepfather] in 1057 as LULACH "the Simple" King of Scotland, crowned Aug 1057 at Scone Abbey, Perthshire. The Annals of Ulster record that "Lulach son of Gilla Comgain, over-king of Scotland was killed in battle by Mael Coluim son of Donnchad" in 1058.](Cawley's Medlands)
m ---. Finnghuala of Angus? The name of Lulach´s wife is not known.
Lulach & his wife had two children: (Cawley's Medlands)
(a) MAELSNECHTAI (-1085). A grant by "Maelsnecte son of Luloeg" to the church of Deer is recalled in a notice of grants between 565 and 1100. Mormaer of Moray until 1078. Monk. (Cawley's Medlands)
(b) daughter. Her parentage is confirmed by the Annals of Ulster which record the death in 1130 of "Angus son of the daughter of Lulach” in a battle “between the men of Scotland and the men of Moray”. The name of her husband is not known.
m ---. One child: (Cawley's Medlands)
(1) ANGUS (-killed in battle Strickathrow 1130). Mormaer of Moray. Moray was conquered by the Scots in 1130 after Mormaer Angus's defeat at Strickathrow. The Annals of Ulster record the death in 1130 of "Angus son of the daughter of Lulach” in a battle “between the men of Scotland and the men of Moray”. John of Fordun´s Scotichronicon (Continuator) records that "Malcolmus filius Macheth" lied to claim he was "filium Angusii comitis Moraviæ" who was killed "tempore…regis David…apud Strucathroth a Scotis". (Cawley's Medlands)