In the video, two men fight one another, with one man seen punching another in the head.
Another fan intervenes and appears to be trying to keep the men apart.
But the bloodied fan instead clambers over the seats, getting closer to the fight. The incident is believed to have been an isolated one.
The match kicked off in Wembley stadium at 7.45pm but Scottish fans had been making the most of their day in the capital, storming through Trafalgar Square and even blocking Covent Garden.
The Tartan Army set up base at Trafalgar Square - which hosted the annual Silence In The Square event earlier in the day.
The Scottish FA pleaded with Tartan Army fans to be respectful of commemorations this morning, with the travelling Scots immaculately preserving silences across the capital.
But it changed later in the afternoon, with attentions turning towards the crunch match with England - while one fan stripped off and jumped into the fountain still full of paper poppies from the morning's events.
Blood pouring from one fan's leg covered a Saltire flag as paramedics attended a concussion, taking at least one fan away in an ambulance.
Public toilets had to be closed due to deliberate damage such as equipment being kicked and tiles being pulled off, police and workers said.
Hundreds of fans blocked Covent Garden as they made their way through the capital for the evening's game.
There were chants of 'Big Sam's a liar, Scotland's gonna qualify' as the fans drank in pubs and in Trafalgar Square.
As the fans made their way to the stadium, they did so to the tune of bagpipes on the tubes.
Scores of people came down from the column, leaving a sea of shattered glass, wet cardboard and spilled alcohol on the landmark.
One of the lions flanking Nelson's column had a traffic cone placed on its head mirroring the Duke of Wellington statue in Glasgow.
Passer-by Nicoletta Morelli said: "It's not so nice to see London like this. I have not forgotten the hooliganism of British fans in Rome."
Gillian Johnson, a heritage warden with the GLA, said it was "terrible", adding: "It's really not fair, this is supposed to be a public place."
A foreman cleaning the mess said he had come to "expect this level of mess" when football fans are in town.
By 6pm the scene was empty as fans started to make their way to Wembley.
Men with large brushes swept away the debris as a strong smell of stale alcohol lingered in the air.
Empty bottles lined the pool rims of the two fountains, which had been switched off after they were filled with fairy liquid by Scottish fans three years ago.
There were a total of two arrests today, one on suspicion of common assault and the other on suspicion of being drunk and disorderly, the Met Police said.