SSO no longer works with VMware vSphere Client (SEC_ERROR_INADEQUATE_KEY_USAGE)

Last week I noticed that the Single Sign-On (SSO) for the vSphere Client (Flex and HTML5) no longer works in my Firefox. Normally, the VMware Enhanced Authentication Plugin toolbar disappears at the bottom and you can enable the "Using Windows Session Authentication" option, but the checkbox remains unchecked. Reinstalling the VMware Enhanced Authentication plugin, updating the vCenter Server and reinstalling the plugin does not work.

Then I open a ticket at Vmware Support. Hours and some technology later, we had no idea what was going on. But, we find out that the local web server at https://vmware-plugin:8094/ (used by the SSO) displays the following error message in Firefox:

SEC_ERROR_INADEQUATE_KEY_USE

It looks like a problem with the Enhanced Authentication Plugin certificate. This is provided by the plugin. It creates a local web server to communicate with the web page. The VMware support team then created the certificate manually, but the error still occurred - even with IE and Edge.

Then I tried it with a fresh portable Firefox and it worked. In my installed Firefox I removed certificate exceptions for the normal host from the vCenter and vmware-plugin. I also - and most importantly - remove the certificate from the vCenter host and the vmware plugin from the certification authorities in Firefox. Reload the page and it's working again.