His real name, Nelson Green. You can put his alias under other names, if you like.
The pension index cards are a great source for aliases. Here's a summary of them (with possible citations):
'U.S., Civil War Pension Index: General Index to Pension Files, 1861-1934', Nelson Green alias Lyman Bailey (A 7 NJ Inf); image copy, Ancestry (https://www.ancestry.com/discoveryui-content/view/1498334:4654 : accessed 31 October 2023); citing Washington, D.C., National Archives and Records Administration, Record Group 15; NARA microfilm series T288.
Lyman Bailey (alias)
Nelson Green
widow Mary J Green
service A 7 N.J. Inf
widow's pension application filed 13 Sep 1865, application no. 109219, certificate 79652
And here's the one on Fold3:
'US, Civil War Pensions Index, 1861-1900', Nelson Green alias Lyman Bailey (A 7 N. J. Inf.); image copy, Fold3 (https://www.fold3.com/image/3194014/bailey-lyman-us-civil-war-pensions-index-1861-1900 : accessed 31 October 2023); citing Washington, D.C., National Archives and Records Administration, 'Organization Index to Pension Files of Veterans Who Served Between 1861 and 1900, compiled 1949 - 1949, documenting the period 1861 - 1942', Record Group 15; NARA microfilm series T289.
Lyman Bailey (alias)
Nelson Green
Private, co. A, 7 Reg't N. J. Inf.
widow's pension application filed 13 September 1865, application 109,219, law [blank], certificate 79,652
died [blank]