authorship - As a postgrad student, is it appropriate not to add my supervisor's name on publication?

Let me summarize the complicated state of affairs, as follows:

This summer, I was cooperating with the other team in a spin off project, we decided to publish the results, so I prepared the paper.

In the authorship, I added my supervisor's name. I think it is usual because I am his student and whatever I've learned came from him.

But when the other team's supervisor saw this, he asked me to drop my supervisor's name from the list. I sense that this is not only because he is not involved in the project, but also because of some political issues.

Dropping my supervisor's name makes me feel bad, like betraying him.