Kate, if I am correct then the Reno Sparks Indian Colony is indeed a party to the litigation, however the Fort McDermitt Paiute and Shoshone Tribes is not, just a group of members of that second tribal group.

I believe the article linked below substantiates that view:

Judge hears fight over lithium mine on Nevada-Oregon line (bigrapidsnews.com)

"U.S. District Judge Miranda Du also agreed to an expedited review of a new request from a Nevada tribe to join the legal battle and seek a similar restraining order based on its claim the mine would disturb sacred burial grounds near the Nevada-Oregon line."  In my view, that "Nevada tribe to join the legal battle.." is the Reno Sparks Indian Colony and not the Fort McDermitt Paiute and Shoshone Tribes (Full Name:  " Fort McDermitt Paiute and Shoshone Tribes of the Fort McDermitt Indian Reservation")

There is no question that a group of members of the Fort McDermitt Paiute and Shoshone Tribes have joined the protest against the development of the LAC Thacker Passs Lithium mine development, but I see no evidence that the Fort McDermitt Paiute and Shoshone Tribes, per se, have become joint litigants.  

That being said, there is no question that the tribal government of the Fort McDermitt Paiute and Shoshone Tribes has become involved in the controversy as evidenced by the fact that  " In April, in response to a petition from tribal members, the Fort McDermitt Paiute and Shoshone tribal council voted unanimously to withdraw the tribe from an earlier agreement with Lithium America under which the tribe would have consulted with the mining company."

In addition: "In order to have their voices heard, Indigeneous protesters formed a group called Atsa koodakuh wyh Nuwu, which translates to “People of Red Mountain.” People of Red Mountain is a group of elders, middle-aged people, and youth tribal descendants from Fort McDermitt who are collectively standing up against this lithium mining project, as well as other proposed mining projects around the Fort McDermitt reservation."

Keep in mind that Judge Du is only ruling on the request for a temporary injunction agaisnt the Thacker Pass construction on 29 July, the "big show" will be later in early 2022 when she gives her legal opine on the merits of the conservationist's overall lawsuit before the end of January, 2022.  So regardless of how the pregame activities play out here in July we still have the pain of enduring until the end of January next year to see how this overall drama will come to a conclusion.

Below are some links to current articles from various sources.  I could be mistaken, but I believe that one of the articles is way, way misleading about the size of the actual proposed mining pit by Nevada Lithium/LAC out at Thacker Pass.  Perhaps someone who has reid the EIS in close detail will jump in here and contribute their knowledge of the actual dimensions of the proposed mine pit, but on first reading of the article I believe the journalist has made a gross exageration.

I apologize for the gray shading of much of this post.  I don't have a clue as to how that happened, but obviously I have interspaced my own comments with designated quotes from the articles linked below and perhaps in doing so I have somehow activated the gray shading in the process.  

Judge hears fight over lithium mine on Nevada-Oregon line (570news.com)

Ranchers, Environmentalists and Indigenous People Oppose Lithium Mine On Sacred Land | The Mick Hitchcock, Ph.D., Project for Visualizing Science (hitchcockproject.org)

Judge hears fight over lithium mine on Nevada-Oregon line (bigrapidsnews.com)

Okiedo