I will assume Alexa is actually hearing you correctly, so try the following…

Try changing the name of your Arc so that it  is ‘unique’ to Alexa and not the same name as the lights ‘room’ name.. you could call it ‘Lounge’ for example, or alternatively you could rename your lights room/group. 
 

In addition ensure your lights and TV are listed in differently named ‘Groups’ within your Amazon Alexa App, or remove your LG TV from the same group and leave it ‘floating’ outside of an Alexa group altogether - the choice is yours, but I think separate groups are perhaps the better option. 
 

Hopefully the above should fix your issue.

A good ‘rule of thumb’ with Amazon Alexa is to have unique ‘device’, ‘room’ and ‘group’ names and you should then hopefully find that the overall control of the smart-home linked devices and their skills will become ‘less confused’ with the intended syntax.

I find the problem, when I add the Alexa enabled arc to Alexa , it creates a virtual “arc tv”, and this is controlled by the arc. Now, when I pull both the arc and the “arc tv” outside of the “living room” group, and I say turn off “living room”, tv is unaffected, which is ok and expected.

however, when the tv is off and I say Alexa turn off “arc tv”, the tv state is flipped and now tv is on. 
 

Seems to me like a Sonos problem where anytime a turn off command is sent to the arc speaker controlled tv, the off command doesn’t turn “off” but just flips the current state” (on>off , off>on)

note: command to turn on works correctly, if tv is on it will stay on.

@Sonos any input?