Sola seems motivated by the same thing:
love for Lancer (there's a lot in the novel that portrays her as a cold, frozen person who had resigned to be a tool to her family, she had no expectations of love or freedom by her upbringing, so she was almost dead inside, then Lancer appeared and she felt alive and unfrozen for the
first time) and she reacts when Kayneth began to
badmouths him all the time (and Lancer stands up for him. It's hilarious). If Kayneth dies, there's a chance Lancer will follow. She
is truly in love with him.
The
three of them are a messy team because of their emotional motivations.
Lancer's love (loyalty) to Kayneth and mistrust on Sola's motivations, he rejects her as Master as we have seen (aka he doesn't form a contract). As much as I feel bad for him, his actions are a bit on the stupid side...
Kayneth is the same. He clearly knows he can't win like this and what she's saying is sensible, but his jealousy blinded him, like his paranoia towards Lancer. Take a look at this:
Spoiler for novel reasoning:
Sola's rank as a magus is severely inferior to Kayneth. However, Waver who summoned Alexander, and the bloodthirsty murderer who formed a contract with Caster; those whose participations as Masters were completely out of the question, were still in this Holy Grail war. As for battle tactics, even for Sola, it is not entirely impossible for her to win through the battles.
And then, when it comes to using a Servant, Command Spells which make them submit to the Master are indispensable. But still...
Kayneth remembered. The hot look Sola had whilst gazing at Lancer had at the hotel late at night after his first battle ended. An intoxicated glance as if she was dreaming, something not shown to him, her fiance, before.
If she was merely fascinated by a handsome man, then that is still excusable. It is just a small, problematic indisposition a woman has. Her admiration towards man to that extent was something which he, as her husband, could not help with.
Yet, Lancer was said to be a case of “not just an ordinary handsome guy”.
I don't need to bring up Sola's actions since she's been already criticized to death here. I agree her motivations were wrong, but I don't think Kayneth would have given her the seals otherwise (he childishly refused even if he understood it was logical). Because his motivations were wrong too. Wrong in the context of the war that's building around them.
It's not so black and white. The three of them are being blinded by the love triangle stuff. It's not only Sola's fault. They should have included a therapist in their team.