Both have been under investigation as part of Operation Branchform, the long-running police probe into allegations that more than £666,000 raised by the party specifically to campaign for independence in a proposed second Scottish independence referendum was spent on other activities.
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Mr Murrell was charged last year “in connection with the embezzlement of funds” from the SNP in April last year.
A month later, the force sent a standard prosecution report to prosecutors concerning incidents said to have occurred between 2016 and 2023.
Police have made a formal request to the Crown Office asking what action to take next with Ms Sturgeon.
In a statement posted on Instagram, Ms Sturgeon said: “With a heavy heart I am confirming that Peter and I have decided to end our marriage.”
The former first minister added: “To all intents and purposes we have been separated for some time now and feel it is time to bring others up to speed with where we are.
“It goes without saying that we still care deeply for each other and always will.”
She added that they would be making no further comment on the matter.
Asked about divorce after a speech to party activists and journalists in Edinburgh, Mr Findlay said: “You know, it’s obviously a private matter.”
He added: “I’ve got a calculator on the phone that’s perhaps less calculating than the former first minister. She doesn’t do very much at all without a reason. I guess it may well be connected to ongoing issues that are happening elsewhere. I guess time will soon tell.”
After her arrest in June 2023, Ms Sturgeon protested her innocence to the media. “I am absolutely certain that I have done nothing wrong,” she said.
Ms Sturgeon’s spokesperson declined to comment on Mr Findlay’s remarks.