In such scenario, you will need to bind the CommandBindingCollection object to the element which implements the command.

In your attached sample, the element which has the minimize and close command is UXWindow so you will need to bind the CommandBindingCollectionto UXWindow. Here is the snippet:

public class SymphonyUXDesktop : UXDesktop
{
    protected override DependencyObject GetContainerForItemOverride()
    {
        UXWindow newWindow = new UXWindow();
        newWindow.IsClientVisible = true;
        newWindow.CanMaximize = true;
        newWindow.CanMinimize = true;
        newWindow.CanMove = true;
        newWindow.CanResize = true;

        CommandBinding windowCloseCommandBinding = new CommandBinding(WindowCommands.Close, HandleWindowCommand, CanHandleWindowCommand);
        CommandBinding windowMinimizeCommandBinding = new CommandBinding(WindowCommands.Minimize, HandleWindowCommand, CanHandleWindowCommand);

        CommandBindingCollection bindingCollection = new CommandBindingCollection();

        bindingCollection.Add(windowCloseCommandBinding);
        bindingCollection.Add(windowMinimizeCommandBinding);

        CommandManager.SetCommandBindings(newWindow, bindingCollection);

        return newWindow;
    }

    private void HandleWindowCommand(object sender, ExecutedRoutedEventArgs e)
    {

    }

    private void CanHandleWindowCommand(object sender, CanExecuteRoutedEventArgs e)
    {
        e.CanExecute = true;
    }
}