How to get the ListBoxItem's default Item/Content/DataTemplate?

Question

I wanted to use the ListBox's default ItemTemplate/DataTemplate/ContentTemplate or whatever it is to reduce some code of my ComboBox like Custom FrameworkElement:

In the Popup I've a ListBox named list and when I select an item from the list it's displayed in a ContentControl named selectedContent. To set the content style of the list as well as selectedContent I've two normal properties, DisplayPath and ItemTemplate:

string display;  
public string DisplayPath {  
    get { return display; }  
    set {   
        display =   
        list.DisplayMemberPath = value;  
        selectedContent.SetBinding(ContentControl.ContentProperty, new Binding() {  
            Path = new PropertyPath($"{nameof(list.SelectedItem)}.{DisplayPath}"),  
            Source = list  
        });  
    }  
}  
DataTemplate itemTemplate;  
public DataTemplate ItemTemplate {  
    get { return itemTemplate; }  
    set {  
        itemTemplate =  
        list.ItemTemplate =  
        selectedContent.ContentTemplate = value;  
        selectedContent.SetBinding(ContentControl.ContentProperty, new Binding() {  
            Path = new PropertyPath(nameof(list.SelectedItem)),  
            Source = list  
        });  
    }  
}   

and in both of these I'm binding the ContentControl.ContentProperty in two different ways. I wanted to move the binding part in the Loaded handler to make those auto properties look like this:

string display;  
public string DisplayPath {  
    get { return display; }  
    set { display = list.DisplayMemberPath = value;  }  
}  
DataTemplate itemTemplate;  
public DataTemplate ItemTemplate {  
    get { return itemTemplate; }  
    set { itemTemplate = list.ItemTemplate = value }  
}   

and the Loaded handler onLoaded like this:

void onLoaded(object sender, RoutedEventArgs e) {  
    selectedContent.ContentTemplate = list.WhateverTemplateItGets;  
    selectedContent.SetBinding(ContentControl.ContentProperty, new Binding() {  
        Path = new PropertyPath(nameof(list.SelectedItem)),  
        Source = list  
    });  
}  

BUT I don't know yet what template is being used by the list so in the onLoaded I currently have to have these to make it work:

void onLoaded(object sender, RoutedEventArgs e) {  
    var binding = new Binding() { Source = list };  
    if (ItemTemplate != null) {  
        selectedContent.ContentTemplate = list.ItemTemplate;  
        binding.Path = new PropertyPath(nameof(list.SelectedItem));  
    }  
    else binding.Path = new PropertyPath($"{nameof(list.SelectedItem)}.{DisplayPath}");  
    selectedContent.SetBinding(ContentControl.ContentProperty, binding);  
}  

can it be simplified by using the default template list is using?