Here are 10 points of friction that have led to serious political sparks:
The BJP is worried about some of its state legislators openly voicing support for Mr Kumar. Earlier this week, the party suspended an MLA, Amarnath Gami, who said BJP led by Sushil Modi has become autocratic.
The BJP boycotted Mr Kumar's trust vote on June 19 after the alliance ended. Sources say that was to avoid the embarrassment of having some BJP legislators vote in favour of Mr Kumar.
On July 7, 10 bombs exploded at Bodh Gaya in Bihar. When the chief minister arrived there four hours after the terror attack, he was greeted by a group of BJP leaders shouting slogans against him. The cries got louder when BJP leader Sushil Kumar Modi reached there. He was Deputy Chief Minister in the BJP-JDU government.
The next day, the BJP supported a bandh called by Lalu Yadav, former chief minister and head of the Rashtriya Janata Dal or RJD.
Lalu, who has served as chief minister of Bihar, has for years taunted Mr Kumar for remaining aligned with the "communal forces" of the BJP.
Narendra Modi's critics including Nitish and the JDU blame him for not doing enough to stop the riots of 2002 in which hundreds of Muslims were killed in Bihar. They accuse Mr Modi of being a divisive leader. The BJP points out that for many years, the JDU was able to swallow its discomfort with Mr Modi. But Nitish's party says that by making Mr Modi campaign-in-charge, the BJP has indicated it's likely to pick the Gujarat Chief Minister as its candidate for Prime Minister.
Nitish has been accusing the BJP and Lalu of a growing political proximity. Sources say that he was irked a few months ago because when Lalu's car met with an accident in Bihar, Mr Modi called to check on him, and then noted the phone call on Twitter.
Now, the JDU has alleged that the BJP and Lalu are teaming up to extract political mileage from the tragedy that left 23 children dead in Bihar last week after they were poisoned by a free mid-day meal at their school. BJP never highlighted the role of RJD leaders in this, instead they kept on attacking Nitish.
Famous lawyer Ram Jethmalani, who was expelled from the BJP, offered recently in public to forge unity between Lalu and Mr Modi. Mr Jethmalani represents both Lalu and BJP General Secretary Amit Shah a close aide of Modi in legal cases they face. His remark was rejected by the BJP but created a storm in Bihar political circles.(Neither the BJP nor the RJP is yet to reject his statement officially)
By repeatedly stressing a nexus between Lalu and the BJP, Nitish hopes to cannibalize their vote banks. The BJP draws heavily upon upper case support in Bihar; traditionally, this audience has been suspicious of Lalu. Nitish hopes they will reject the BJP because of a perceived proximity to Lalu, and turn to him in the national elections that are due by May. Similarly, Muslims who are important for Lalu's party will be concerned about any proximity with the BJP.
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