How do I register a UCLA Domain Name?

My department chair asked me to come up with a short, catchy, stand-out domain name, how do I know whether I can do this and, if so, how do I go about registering it?

UCLA Policy 411 deals with the registration and use of UCLA domain names and can be found at: http://www.adminvc.ucla.edu/appm/public/app_0411_0.html

Once you understand the policies that pertain to your request, the next step is
Domain Name Registration"