Android icon showing when turning Wi-Fi on
I have an LG phone aswell (not the G4) and icon that you are described is probably icon which shows up when phone is checking for updates of android or updates for LG apps. You can probably change when it will check for updates. I don't have any sources about this but I know that  because I catch notification several times when on slow cellular data.
Edit here is the photo of icon I get when checking for updates:


Also here is "app updater" app icon it's very similar to picture you posted