The Regional Independent Group and Government parties will not challenge Ceann Comhairle Verona Murphy's ruling that the group's four TDs cannot form a Dáil technical group.
However, the group will seek a significant reform of the Dáil's standing orders to ensure that Independent TDs supporting the Government get speaking rights in the chamber.
The Regional Group has conceded that they will now not be part of a Dáil technical group which aimed to position itself on the opposition benches, while supporting the Government.
But the group wants some solution found to give Danny Healy-Rae, Barry Heneghan, Gillian Toole and Michael Lowry speaking time as they believe they are now caught in no man's land.
Mr Lowry said the Government now has to examine options to accommodate what he termed supportive Independents.
The voice of all duly elected Independents must be heard and a framework put in place, he said.
A spokesperson on behalf of the Taoiseach and Tánaiste said that the two Government parties accept the Ceann Comhairle's recommendation, and that Government Chief Whip Mary Butler will "continue to engage with the Dáil reform committee to ensure speaking rights for all members are upheld."
Speaking ahead of today's meeting of the Dáil reform committee, Sinn Féin's Padraig Mac Lochlainn and Social Democrats Cian O'Callaghan urged the Government to accept the Ceann Comhairle's ruling.
Government Chief Whip Mary Butler has said work will take place over the coming weeks to reform standing orders which will ensure that all TDs in the Dáil will be able to access speaking time.
She said the Dáil can resume tomorrow and begin addressing issues that the country wants their public representatives to be debating including the aftermath of Storm Éowyn.
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Speaking on a visit to Mayo University Hospital in Castlebar, Ms Carroll MacNeill said the Ceann Comhairle had gone through an important process, taking legal advice, reflecting on it and making a determination.
"She is the office holder, she has gone through her own process and that's her decision," she said.
Ms Carroll MacNeill said she felt that work now needed to be done "to make sure that everyone in the Dáil, who has the same mandate given to them by their communities" had an opportunity to speak in the chamber.
The minister said that included government backbenchers.
Ms Murphy reached her conclusion having considered legal advice, submissions and statements made by the TDs themselves.
Essentially the Ceann Comhairle was not satisfied that Mr Lowry, Ms Toole, Mr Heneghan and Danny Healy-Rae are members "in Opposition" as required by the rules of the Dáil.
Read more: Ceann Comhairle's Dáil row move sends shockwaves through Govt
Regional Independent Mattie McGrath described the ruling as unbelievable, adding that it was 'GUBU' territory.
The ruling was welcomed by Mr Mac Lochlainn, who said it was clear that you cannot be in opposition and government at the same time and he also contended that it was deeply embarrassing for the Government.
Labour Leader Ivana Bacik also welcomed the decision saying it vindicated the stance taken by opposition parties.
Mr O'Callaghan said the Ceann Comhairle put great weight behind the words of members of the Regional Independents, when the Government was formed.
