If "Shinzaemonnojo NAGAI" that appears in the documents from the Daiei era (1521-27) is the same person as Dosan's father, then this implies that Dosan's father was already using the name NAGAI.
大永年間の文書に見える「長井新左衛門尉」が道三の父と同一人物であれば、既に父の代に長井氏として活動していたことになる。