POWER OF ATTORNEY

              KNOW ALL BY THESE PRESENTS, that the undersigned hereby
constitutes and appoints each of Debra Bradford, Thomas A. Sa and Joey Tongson,
each with full power to act singly, as the undersigned's true and lawful
attorney-in-fact, with full power of substitution, to:

1)       execute for and on behalf of the undersigned, in the undersigned's
         capacity as an officer and/or director and/or beneficial owner of
         California BanCorp (the "Company"), an SEC Form ID and Forms 3, 4, and
         5 in accordance with Section 16(a) of the Securities Exchange Act of
         1934, as amended, and the rules thereunder;
2)       do and perform any and all acts for and on behalf of the undersigned
         which may be necessary or desirable to complete and execute any such
         Forms ID, 3, 4, or 5, complete and execute any amendment or amendments
         thereto, and timely file such form with the United States Securities
         and Exchange Commission and any stock exchange or similar authority;
         and
3)       take any other action of any type whatsoever in connection with the
         foregoing which, in the opinion of such attorney-in-fact, may be of
         benefit to, in the best interest of, or legally required by, the
         undersigned, it being understood that the documents executed by such
         attorney-in-fact on behalf of the undersigned pursuant to this Power of
         Attorney shall be in such form and shall contain such terms and
         conditions as such attorney-in-fact may approve in such attorney-in-
         fact's discretion.

              The undersigned hereby grants to the attorney-in-fact full power
and authority to do and perform any and every act and thing whatsoever
requisite, necessary or proper to be done in the exercise of any of the rights
and powers herein granted, as fully to all intents and purposes as the
undersigned might or could do if personally present with full power of
substitution or revocation, hereby ratifying and confirming all that such
attorney-in-fact or such attorney-in-fact's substitute or substitutes, shall
lawfully do or cause to be done by virtue of this power of attorney and the
rights and powers herein granted.

              This power of attorney shall remain in full force and effect until
revoked by the undersigned in a signed writing delivered to the attorney-in-
fact.

              IN WITNESS WHEREOF, the undersigned has caused this Power of
Attorney to be executed as of this 2nd day of August, 2021.

                                    By:   /s/ Millicent C. Tracey
                                          ----------------------------
                                    Name: Millicent C. Tracey