The description of the component should be rewritten.
ContextMenu: attachTo attribute
"attachTiming" attribute was added
Key features were added
Task is in progress.
Exploring related tasks.
Exploring <rich:componentControl> description, given in Developer Guide, and it work via LiveDemo.
Exploring components with similar functionality: <a4j:jsFunction> and <a4j:support>.
So, the task is done. I rewrote just part of "Details of usage" from the beginning to words "With the help of the 'attachTiming'"