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Kinship
There are three versions. The Japanese dictionaries and Markus Sesko make him a son-in-law of Rai Kunitoshi; Sesko also gives him as an older cousin of Rai Kunimitsu; the Japanese Wikipedia, on the other hand, records that he was a son or student of Kunimitsu. Sesko notes that the first two may both be true, given the custom of adoption; the third does not fit with them.
Dates
His few dated works run from the Karyaku era (1326–1329) to the Shōkyō era (1332–1334). He was a contemporary of Kunimitsu and, according to Sesko, helped Kunitoshi in his last years by signing on his behalf (代銘, daimei). The 『古刀銘尽大全』 (Kotō Meizukushi Taizen), a late and unreliable book, has him born in 1247 and dying in 1324, and says that he went to Kamakura in 1274 and returned to Kyoto in 1302. Those dates do not fit those of his blades, which are later than the death the book gives him.
A student of Masamune?
The Edo sword books place him among the ten students of Masamune (正宗十哲, Masamune jittetsu). Sesko leaves two readings open: that he went to Kamakura in person to learn the Sagami technique, or that, without leaving Kyoto, he adapted his style to the fashion of the moment. The nickname “Kamakura-Rai” fits both, like “Kamakura-Ichimonji” for Sukezane of Bizen; at the time people said “Kamakura”, not “Sōshū”. A phrase that Sesko quotes sums it up: he was influenced by Sōshū, but he did not make sōshū swords. He kept the forging of the Rai school and let the Sagami influence reach above all the hamon.
Lineage
He is followed by Rai Kunihide (来国秀, Rai Kunihide), who is held to be his son, and Rai Hidetsugu (来秀次, Rai Hidetsugu). Some take them for one and the same smith and for a second Kunitsugu; Sesko treats them as two different smiths.