The Senate has voted 35-21 to note that its newest member, Victorian Labor senator Kimberley Kitching, was found to have provided untruthful evidence to the Fair Work Commission (FWC).

In a rare united stand, the Greens joined the Coalition in backing the motion, which also received support from three One Nation senators and Victorian senator Derryn Hinch. One Nation’s Rod Culleton, who earlier represented himself at a High Court directions hearing about his eligibility to be in parliament, did not vote but his office said he had not deliberately abstained.

The Nick Xenophon Team did not take a stand. Liberal Democrat David Leyonhjelm also abstained. Tasmanian Jacqui Lambie voted with Labor against the motion.

The choice of Kitching to replace former Victorian senator Stephen Conroy was controversial within Labor, dividing the right faction. Her preselection was backed by Opposition Leader Bill Shorten, a long-time supporter. The government, including Prime Minister Malcolm Turnbull, has been taking cracks at her since her arrival a few weeks ago.

The Senate motion was moved by former employment minister Eric Abetz. It said the Senate noted that the FWC found Kitching, a former official of the Health Services Union, gave “untruthful and unreliable evidence”. “It also found her conduct strikes at the heart of the integrity of the right of entry permit system,” the motion said.

The motion called “on anyone appearing before the Fair Work Commission to provide reliable and truthful evidence”.

The Heydon royal commission on trade unions recommended Kitching be prosecuted for undertaking tests on behalf of other officials to obtain right-of-entry permits. But no action has been taken against her.

Abetz said Labor’s “protection racket for dodgy union bosses” was on display. “Their refusal to call on anyone appearing before the Fair Work Commission to provide reliable and truthful evidence undermines the rule of law in Australia,” he said.

“A finding was made against Ms Kitching by a senior member of the commission. That finding was then upheld on appeal by three members of the commission. A year later, we are then told by Labor that Ms Kitching represents the gold standard which should be in the Senate,” Abetz said.