You will have to customize the code to do that.
The product-details for the email are built in the create_add_products() function inside the order.php class, and the actual email is sent by the send_order_email() function of that same class.
The admin copy of the email includes the customer's email, sandwiched between the auth-code at the top and some office-use-only info at the bottom.
To extract the product details for just the admin email will require some duplication of code and reworking of the overall content. Be sure to test that you haven't fudged the customers' email after making your changes.