A further two employees have been disciplined following inappropiate behaviour on the websites, the BBC has disclosed under a Freedom of Information request.
The news comes after the BBC's acting director of news, Fran Unsworth, last week sent an email effectively banning staff from using Twitter.
Although the report did not name the Tory peer, he was widely identified online, including on Twitter.
The BBC's Acceptable Use Policy reads: "Your use of social networking sites such as Facebook must not interfere with BBC work. The BBC reserves the right to block access to any internet site that is interfering with the operation of normal BBC business."
However, Steven George-Hilley, director of technology at right-wing think tank Parliament Street, which obtained the figures, said organisations should train staff so that they do not publicly tweet grievances in the first place.
"Misuse of social networking sites such as Twitter and Facebook by employees can compromise the integrity of publicly funded organisations and trigger long-term reputational damage.
"Simply writing up an acceptable use rulebook is not enough. It is vital that staff are trained to fully understand the consequences of their actions to prevent these incidents from occurring in the first place."
The BBC has declined to comment on the story.
