Having been axed on Monday, Arthur was en route to South Africa via Dubai after hearing that his mother, Lynne, had breathed her last.
"It's not a nice time for us as a family, but you know, that happens as well," he told reporters.
Arthur's mother died in Kimberley, in the Northern Cape Province of South Africa. She had been ill for many months.
The 45-year-old kept his composure at his departure press conference despite feeling gutted about being axed as coach, taking responsibility for the team's inconsistent results, the most recent being the Champions Trophy debacle.
"I was really, really shocked," Arthur said.
"I know we have had a few issues but I felt the team was moving in the right direction. I was confident in the team we had and felt the team discipline issue was getting there.
"I am gutted. I didn't see it coming. I can look back at it and say I've chucked everything at it. I think I leave with a lot of professional pride, I think I leave with a lot of professional dignity," he added.
Everyone from Cricket Australia chief James Sutherland to captain Michael Clarke to new coach Darren Lehmann paid tribute to Arthur as a decent man who had invested great time and effort into the team and his players.
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