Right after the debate in the House of Commons, Robert talks to Gertrude in the hall way. Behind them is a statue head. Several shots later that statue head magically moves to the opposite wall where Arthur "talks" to it.
The play that all are watching when Lord Goring fools around with his looking glass and Gertrude asks Robert what his business with Mrs. Cheevley is, is "The Importance Of Being Earnest" by Oscar Wilde. In the end you see the author coming out on the stage and saying a few funny words. This really occurred at the premier, Oscar Wilde did really say these things. It is a kind of play in a play, as "An Ideal Husband" was also written by Wilde.