Couldn't stop thinking "my grandfather's dog yesterday" and I was very confused for a minute
Jag beklagar: People used to say beklagar sorgen, but now many people feel that it's not the grief that is to be lamented, but the loss, so nowadays it's better to say Jag beklagar din förlust. (But Jag beklagar works fine on its own).
That's a very good point! It makes much more sense to be sorry about someone's loss than someone's grief.