I was asked to make flyout tabs open or close with a mouse click. Took a bit of doing so I thought I'd share it here as a way of saying thanks for all the other helpful examples I've taken advantage of.

Here's my page XAML

<Window
  xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
  xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
  xmlns:igDock="http://infragistics.com/DockManager"
  x:Class="FlyoutTabTest.MainWindow"
  x:Name="Window"
  Title="MainWindow"
  Width="640" Height="480">

  <Grid x:Name="LayoutRoot">
    <igDock:XamDockManager x:Name="PART_RibbonDockManager" Theme="Office2k7Silver"
      FlyoutAnimation="Slide" Focusable="False">

      <igDock:ContentPane>
        <StackPanel>
          <TextBlock Text="Hi"/>
        </StackPanel>
      </igDock:ContentPane>
           
      <igDock:XamDockManager.Panes>

        <igDock:SplitPane x:Name="_splitHistoryPane"
          igDock:XamDockManager.InitialLocation="DockedRight" MinWidth="250">
          <igDock:ContentPane x:Name="HistoryPane"
            Header="History" IsPinned="False" AllowClose="False"
            AllowDockingBottom="False" AllowDockingTop="False"
            AllowDockingInTabGroup="False">
            <StackPanel>
              <TextBlock Text="Hi"/>
            </StackPanel>
          </igDock:ContentPane>
        </igDock:SplitPane>

        <igDock:SplitPane x:Name="_splitPresencePane"
          igDock:XamDockManager.InitialLocation="DockedRight" MinWidth="250">
          <igDock:ContentPane x:Name="PresencePane"
            Header="Presence" IsPinned="False" AllowClose="False"
            AllowDockingBottom="False" AllowDockingTop="False"
            AllowDockingInTabGroup="False">
            <StackPanel>
              <TextBlock Text="Hi"/>
            </StackPanel>
          </igDock:ContentPane>
        </igDock:SplitPane>
                   
      </igDock:XamDockManager.Panes>
           
    </igDock:XamDockManager>
  </Grid>
</Window>

And the code-behind:

using System.Windows;
using System.Windows.Input;
using Infragistics.Windows.DockManager;

namespace FlyoutTabTest
{
  /// <summary>
  /// Interaction logic for MainWindow.xaml
  /// </summary>
  public partial class MainWindow : Window
  {
    public MainWindow()
    {
      this.InitializeComponent();

      // Insert code required on object creation below this point.

      // Register preview mouse down events on the flyout tabs
      EventManager.RegisterClassHandler(typeof(PaneTabItem),
        Mouse.PreviewMouseDownEvent,
        new RoutedEventHandler(OnPaneTabItemMouseDown), true);
    }

    // handles preview mouse down events on flyout tab headers to open/close tab
    // when not pinned
    private void OnPaneTabItemMouseDown(object sender, RoutedEventArgs e)
      {
        PaneTabItem pti = sender as PaneTabItem;
        if (pti != null)
        {
          ContentPane pane = pti.Pane as ContentPane;
          if (pane != null)
          {
            // don't mess with pinned content panes
            if (pane.IsPinned == false)
            {
              // if flown out and in an unpinned state, fly back in
              if (pane.PaneLocation == PaneLocation.Unpinned)
              {
                pane.ExecuteCommand(ContentPaneCommands.FlyIn);
              }
              else
              {
                pane.ExecuteCommand(ContentPaneCommands.Flyout);
              }
            }
          }
        }
      }
    }
}

The window initialization registers a class event handler that intercepts PreviewMouseDown events on any PaneTabItem objects. These are the tab headers on an unpinned pane.

When a mouse click occurs, the handler obtains the content pane from the pane tab item. If the content pane is unpinned, it gets tested and either flown in or out.

There's a little quirk if the pane is displayed as a result of a hover rather than a click that I wasn't able to figure out. But maybe this will help somebody else out.