When saving a configuration form, not just actual changes, but everything is saved. Based on the description, this looks like a problem in the unspecified security realm configuration form. Without more details, it's impossible to help you; it's however very unlikely that matrix-auth is the culprit, it's just what motivated you saving the configuration.

Given https://en.wikipedia.org/wiki/Bronco_(disambiguation) reveals no useful technology-related terms: Please note that you're reporting this issue to the Jenkins project. This is not an issue tracker internal to your organization.

While there may be a bug here, it's extremely unlikely to be in matrix-auth, so closing this issue.

Bronco is the name of our Jenkins server, I changed it in the text shortly after it was created.
Where should we put the bug if not on matrix-auth?

 Where should we put the bug if not on matrix-auth?

I don't know, you haven't provided enough information yet. As I mentioned in my first comment, it's not even clear what your security realm is. The screenshot indicates there's a problem with the current configuration already before you submit the form.

I suggest you find someone to assist you with narrowing down the cause of this problem if you don't know how to do that yourself. The forum at community.jenkins.io or Gitter/Matrix chat might be good locations to get started.

It's as simple as it sounds, as soon as I add a group to the matrix, all groups lose their access.
By saving and reloading this page, you can see the error in the image. 
Before adding a new group there was of course no error.
If you then log out, you cannot log in again. Neither can any other user.

Reason for this:
In the Jenkins config.xml not only the added group is added, but also the encrypted password, which establishes the communication with the domain, is changed. As a result, access is lost for all user groups.

Workaround for us:
Add groups manually in config.xml. Do not use the matrix-auth.

It's as simple as it sounds, as soon as I add a group to the matrix, all groups lose their access.
By saving and reloading this page, you can see the error in the image. 
Before adding a new group there was of course no error.
If you then log out, you cannot log in again. Neither can any other user.

Could you confirm that the same problem does not occur when you don't change groups, but still save/submit the configuration form?

What is the config.xml content

• before submission
• after submission (no changes)
• after submission (add a group)

?