Time machine won't backup my external ssd.

I Have a Mac running Big sur with Seagate 4TB backup hub as a Time machine drive.

I recently connected Sandisk extreme SSD to the mac but Time machine won't backup the SSD.

The list of the SSD (shown as WORK in the screenshot) were greyed out.

I have tried unmount and format the Time machine and start over but it still not work.

The content of the WORK is all my work so it's very important to backup them. What should I do?

Thank you in advance for any suggestions and sorry for my bad English.