May 23 (Reuters) - The California State Teachers'

Retirement System has voted against re-electing Exxon Chairman

Darren Woods and Lead Director Joseph Hooley, joining other

public systems concerned the energy company's lawsuit against

climate activists could diminish investor rights.

In a statement sent by a representative, CalSTRS said the

votes were "a measured approach to hold two major leaders of

Exxon accountable to investors." An online record shows CalSTRS

cast votes for the company's 10 other director nominees.

The votes by CalSTRS for Exxon's May 29 annual meeting were

in line with previous comments by one of its board members,

State Treasurer Fiona Ma, expressing concern about the suit.

CalSTRS still took a softer approach than the state's other big

pension system, for public employees, which said it would vote

against the whole board.

Neither system is among Exxon's 40 largest investors, but

they have played influential roles in previous Exxon director

elections.

In January Exxon sued activists to block a vote on a climate

proposal they submitted, sidestepping the usual regulatory

process. Although the investors withdrew their resolution, Exxon

continued the lawsuit, seeking legal costs and other relief.

On Thursday, a U.S. judge allowed Exxon's suit to proceed

against one of the two activist groups it had sued,

Massachusetts-based Arjuna Capital. The judge also said Exxon

could not pursue its claim against Netherlands-based group

Follow This, as it was outside the court's jurisdiction.

Exxon said in a statement the ruling moved it closer to its

goal of reforming the shareholder resolution process. It has

previously said securities regulators have allowed too many

resolutions to come to a vote, costing companies money.

"It's time to stop the abuse of the system, and we're

pleased the judge agreed that we're entitled to our day in

court," Exxon said.

Asked about the CalSTRS votes, an Exxon representative

reiterated its past comments its board has overseen significant

shareholder value creation.