This should have been the Greens best week in Government.

The party succeeded in pushing through its tough and ambitious Climate Action Bill but that win has been overshadowed by the bitter row over Dublin Lord Mayor Hazel Chu's independent run for the Seanad.

The move has deepened divisions and exposed the faultlines in the party, with some elected members struggling with the compromises necessary to push through core policy in a three-pronged coalition.

Ms Chu insists this is about the importance of leading by example in running more women from diverse backgrounds. But others have questioned the sincerity of this motivation by choosing to enter a contest she admits she probably won't win in a move guaranteed to be disruptive.

One parliamentary party member said: "I don't buy Hazel's rationale – I don't think she understands or cares how difficult this could make it for us in Government."

And that point is what elevates this from a typical internal party wrangle – because it threatens to destabilise the Coalition.

However, Ms Chu rejects this, saying that as party chair, she understands the importance of compromise in Government and she says her motivations are genuine.

"Until people see diversity is to be celebrated, we are not going to fix things. I have had protesters at my door and my daughter has been called a mongrel, I don't think people understand as they're not in the same shoes."

However, others see her run as incompatible with being the chair of the party. She sought and got the backing of six members of her own parliamentary party and others to be added to the ballot. If elected, she says she will take the Green whip.

One of those who signed her nomination papers is deputy leader Catherine Martin, putting her on a collision course with leader Eamon Ryan who made it that the party should back the Fianna Fáil and Fine Gael candidates in the poll.

There's disagreement in the party over whether this amounted to a formal pact. Chu's camp say there is no formal agreement but others say there is a tacit understanding that the Green Party will get a quid pro quo should another Seanad vacancy come up within the lifetime of the Government.

The upshot is a deeply split party with two entrenched opposing factions - nominally led by Mr Ryan and Ms Martin.This once again has exposed the lack of party unity.

Ms Martin revealed her ambition by running against Mr Ryan in last year's leadership election. The shock was that the result was tighter than expected - with fewer than 50 votes between them.

Many felt the issue was not completely settled and it's clearly bubbling under. Asked this week if she had confidence in Eamon Ryan as party leader, Ms Chu said she did but she added that he had been in the role for a long time.

Some have also hinted that Mr Ryan has been focused primarily on the Climate Bill and while he is doing good work, the operations of the party has been neglected.

"This needs leadership, Catherine has been suggesting compromises. Eamon is very busy in his department, now that the Climate Bill is done, he can come back."

Others believe Catherine Martin has not been giving the party leader the support he would expect. "To be constantly trying to chop Eamon's ankles off is horrendous."

From speaking to parliamentary party members this weekend, it's clear the schism runs deep and that the disagreement over going into Government with Fianna Fáil and Fine Gael has not been resolved.

"There's a rump of us who didn't want to go into Government and we have never been accepted, we raise concerns at meetings and we are not listened to. We're told we have to have unity but it has to be unity on their terms."

However, another view held by some in the parliamentary party is that others did not want to go into coalition because they believed it would be electorally advantageous to stay in opposition.

The whole issue was further inflamed this week when a parliamentary party meeting dissolved in acrimony after three senators proposed a no-confidence motion sanctioning Ms Chu.

For Chu loyalists, this was a pointless exercise which unnecessarily ignited the situation and had no basis in the party's rules.One Chu supporter was taken aback by the escalation, pointing out that the Greens seem to go to "all-out war" when disagreements happen.

"Just let her run, I want to see people from diverse backgrounds. Let it play out, it's not the end of the world."

For others, the motion lay down a marker, as doing nothing was not an option. "She is doing a solo run and trying to ride two horses as a Green and as an Independent. She could run against a Green and she's overseeing electoral strategy - that's GUBU territory. She has to step down as chair."

That motion remains on the agenda and the parliamentary party will return to it on Wednesday although all sides hope there may be some resolution before then.

Ms Chu insists there is no reason for her to stand aside and she says she has no role in overseeing Seanad elections. "I won't step down until the members want me to step down and a lot of members are encouraging me."

Another factor in all this is Ms Chu's own personal ambition. In the run up to last year's general election contest, she sought to stand for the party - building on her huge local election victory.

The problem was that her area is also Dublin Bay South - the constituency of party leader Eamon Ryan who felt the party could not win two seats. One supporter says: "She feels she is always blocked - the general election, the Seanad selection and now this. The party says it wants to support women but it's never the right time."

Ms Chu says this is not about doing a solo run and rejects the suggestion that there is "personal animosity" between her and the party leader.

"If I wanted a solo run, I would have done so in General Election 2020. I was the good soldier for the good of the party. There is no animosity, this is me wanting to do this."

However despite the negative headlines, all in the party point to the big success with the formal launch of the Climate Action Bill.

"This is deflating but it's so insignificant, the Bill is so important and the other stuff is bull**t - my happiness that the Bill launched far surpasses my annoyance over this, although we would be doing a lot better if this hadn't happened."

And what does all mean for the coalition? The official line from Fianna Fáil and Fine Gael is no comment.  However, any row over the future direction of a government party is destabilising.

One parliamentary party member says the three parties still don't have the relationships and the trust. "Trust in politics is so important, many of us are working on that but this is two fingers to those efforts."

The basic currency of Coalition is supporting others' pet policies and candidates at times. A push against backing the agreed Fianna Fáil and Fine Gael Seanad nominees is a cause of concern for those parties.

The recent Budget was supported by all Government TDs but there's an awareness that future Budgets will be tougher when the massive pandemic spending must be reined in and the public finances stabilised.

It's at that point that coalition loyalty will be sorely tested.