Dan,
Just a thought for natural incoming links:
As a user (taking my SEO glasses off for a moment), if I did like the site - and know nothing of the importance of links, aka.. the everyday users, I would probably link to domain.com and not subdomain.domain.com - unless I came directly off a blog article...
That may be what the company wants: for users to link to the training application (not really sure)? It seems as if they have the www. redirecting to subdomain (so all www.domain.com links would actually hit the subdomain) but everything else would steer clear of the domain.
Hope this helps,
Eli