I found the latest iteration of the trademark guidelines. Basically they don't want "Cal" and "Berkeley" used at the same time. And no mixing of "Berkeley" with any anything related to the athletic department or athletic nicknames.
https://bcbp.berkeley.edu/sites/default/files/TrademarkGuidelinesAndRequirements0102207.pdf
The following general provisions pertain to persons and organizations affiliated with the campus as well as to outside entities and agencies:
a) The campus should not be referred to as "Cal Berkeley." Acceptable nominations are "UC Berkeley," University of California Berkeley," "Berkeley," "Cal," or "California." All variations of the University's name are trademarks of the University of California Berkeley, and property of the UC Regents.
b) It is unacceptable to combine "Berkeley" with "Bears," "Golden Bears," "Athletics," or any athletic team designation (i.e., "Berkeley Football"). The athletic teams should only be referred to as "California Golden Bears," "Cal Bears," or "University of California Golden Bears". The following trademarks may also be used when referring to the University of California programs and/or the campus: "Golden Bears" (the team name), "Go Bears!," "Bear Territory," or "This is Bear Territory." The preceding names and phrases are trademarks of the University of California Berkeley, and property of the UC Regents.
c) "University of California" may not be used without specifically indicating the Berkeley Campus (a reference to the Golden Bears is sufficient to indicate the Berkeley Campus). Examples of correct uses include: "University of California Golden Bears" or "University of California, Berkeley".