The following information taken from Medieval English Genealogy website at:
http://www.medievalgenealogy.org.uk/families/arg/argoutl1.shtml
Giles's son and heir Reginald was said to be aged 40 at his father's death (Calendar of Inquisitions Post Mortem). This is presumably a round figure, and probably means that he was born in the decade before 1242 (or thereabouts).
Reginald distinguished himself by marrying the daughter of an earl - Lora, daughter of Hugh de Vere, the 4th Earl of Oxford. She brought to the marriage the manor of Ketteringham, in Norfolk, which, as was then usual, the Argenteins were to hold as tenants of the de Veres. Ketteringham is known to have come into Reginald's hands between 1262 and 1265, so it is likely that the marriage took place in the period of Simon de Montfort's ascendancy, in 1264 or1265, when Reginald's father Giles was among the baronial leaders. Lora's brother Robert, the then earl, was also a keen supporter of de Montfort (Complete Peerage, vol.10, p.216).
Like his father, Reginald suffered the consequences of Simon de Montfort's defeat at Evesham in 1265, after which his estates were confiscated (Calendar of Inquisitions Miscellaneous) and he was imprisoned. In February 1266, his wife Lora was granted the manor of Ketteringham for the maintenance of herself and her children during his captivity (Patent Roll).
There is little evidence of Reginald's activities in the next few years. He and Lora acquired land at Little Melton (close to Ketteringham), in Norfolk, in 1272 (Feet of Fines). He occurs in connection with Fordham manor around 1274 (Hundred Rolls). At about the same time he is noted among the coheirs of his grandfather Robert de Aguillon (his mother presumably being dead by this time), in entries in the Hundred Rolls concerning various manors in Norfolk.
His father Giles settled certain of the family estates on him in his lifetime: in Melbourn (before 1280) (VCH Cambridgeshire) and in Berton, Suffolk (in 1280-1) (Calendar of Inquisition Post Mortem). But Reginald had not much longer to wait before he succeeded to the main family estates, on Giles's death in 1282. At about the same time, he seems to have disposed of the lands he had inherited from his mother in Norfolk, conveying them to Andrew de Sackville by a fine (Rye). (As discussed elsewhere, this Andrew Sackville has frequently been identified as a son of Reginald's mother Margery, presumably because he succeeded in her estates.)
Earlier, in May 1282, Reginald had been summoned to a muster at Worcester, for military service against the Welsh (Parliamentary Writs). He was similarly summoned to a muster at Montgomery, in May the following year (Parliamentary Writs). Later in the year, Reginald was summoned to the Parliament at Shrewsbury, in September (Parliamentary Writs). (Although his father Giles seems, sporadically, to have occupied a much more influential position among Simon de Montfort's barons, it is this, and a similar summons in 1297 (Parliamentary Writs) which have entitled the family to an account - usually rather brief - in the Peerages. None of the family was ever summoned to Parliament again.)
Reginald's official career continued for the next few years in a fairly low key. He was again summoned, for service against the Welsh, to a military council at Gloucester, in July 1287 (Parliamentary Writs). In August 1295 he was appointed a constable near Dunwich, in Suffolk, for the defence of the coast (Close Roll). The following year he was enrolled, as a knight holding lands in Essex, for the defence of the coast, but was found not to be resident in the county (Parliamentary Writs).
In March 1297, he was appointed a commissioner for the counties of Cambridge and Huntingdon, to enquire into those fostering discord between the king and his subjects (Patent Roll). In the same year, he was summoned to a Parliament at Salisbury in February, to a military council at Rochester in September, and to a muster at Newcastle-upon-Tyne in December, for military service against the Scots. The following year he was summoned again for service against the Scots, to a muster on the king's return to England in January, and to another at York in May. He received his final summons for military service in June 1301 (when he must have been approaching 70), to a muster at Berwick-upon-Tweed (Parliamentary Writs).
Reginald died shortly before 3 March 1308 (Complete Peerage, vol.1), and was buried at Baldock, where he had founded a chapel, and where his gravestone, with a rhyming French inscription, still remains.
Reginald left, in addition to his son and heir John:
a son Richard, on whom he is said to have settled the manor of Acton in Suffolk in 26 Edward I (c.1297) (Argentein evidences).
apparently also a son Giles, who, with the prior of Wymondley, had safe keeping of the charters settling property on the first marriage of Reginald's son John. However, the relationship is stated only in pedigrees.