Tensions between the Houthi and Saleh's party rose Wednesday after Saleh's supporters refused access of the Houthis to the Saleh Mosque in the south of Sanaa to secure a religious ceremony to commemorate birthday of Prophet Muhammad.
The capital Sanaa has been shaken by escalating in-violence this week between supporters of Saleh and the Houthis, with more than 40 fighters killed and injured and residents now fearing a new front in an already devastating war.
Pointing to worldwide law, Lowcock said, "Wars have rules, and they need to be complied with".
Yemeni President Abdrabbuh Mansur Hadi said all those who have identified a clear position in support of the popular uprising aimed at ending the coup and the restoration of the state "will all be partners in the present and future of Yemen". The Houthi allied themselves with the forces loyal to Saleh and have fought a devastating war against forces loyal to Hadi and his foreign backers.
Three Houthi leaders have so far been killed in the clashes that began on Friday, revealed the source, while noting that there have been more casualties among the militia ranks than among Saleh's forces.
The Houthis accused Saleh of betrayal, and vowed to keep up the fight against the Saudi-led coalition.
Mr Hadi, in a statement after a meeting with his advisors, also said he was ready to work with Mr Saleh against the Houthis.More news: Michael Flynn`s actions during the transition `were lawful`: Donald Trump
More news: Padmavati controversy: Sanjay Leela Bhansali, Prasoon Joshi appear before Parliamentary panel
More news: Nissan Kicks Juke Out Of The Range At The LA Auto Show
The statement was issued after a meeting between Hadi and his advisers to form a broad national alliance in the face of the militias Houthi.
"It is not odd or surprising that Saleh turns his back on a partnership he never believed in", the group's political bureau said in a statement.
But the head of the Houthis' Ansarullah group warned that the biggest victor from what he described as Saleh's "sedition" was the Saudi-led coalition.
Abdel-Malek al-Houthi, the leader of the rebels, called on Saleh to "show more wisdom and maturity" in a separate statement.
The United Nations envoy to Yemen, Ismail Ould Cheikh Ahmed, and the ICRC urged the parties to avoid targeting civilians.
Saleh's GPC party accused the Houthis of failing to honor a truce and said in a statement on its website that the Houthis bear responsibility for dragging the country into a civil war.
"The GPC appealed to the army and security forces to remain neutral in the conflict". In 2015 he fled to Saudi Arabia, which used its military force in a bid to place him back in power in Yemen.