Is using alcohol mixed with coloring to paint on cakes unlawful, knowing that its effect and smell disappear after painting?

Alcohol is the intoxicating substance in wine, and "every intoxicant is Khamr, and every Khamr is Haram (forbidden)," and "that which intoxicates in large quantities, a small quantity of it is Haram." Materials containing alcohol are impure (najis) and not permissible to use. Therefore, if a color contains alcohol before it transforms into another substance and loses its intoxicating property, it becomes impure and its use is forbidden. Consequently, if a color mixed with alcohol is added to cake, it renders impure whatever it touches of the cake and makes it forbidden to consume, unlike the rest of the cake that was not mixed with it.