After a sharp spike in fatal construction accidents and other problems at the city Department of Buildings, Mayor Bloomberg said yesterday he was not satisfied with the agency and was examining where mistakes had been made.

Amid a historic building boom in New York City, 13 people have died in construction accidents this year, one more than the total for all of 2007.

The deaths this year include a window installer who fell nine stories last week when his safety strap failed, and the seven people killed last month when a crane collapsed at a site where a 46-story condominium was being built. In that case, a city inspector was arrested and resigned after authorities said he falsely claimed he had inspected the crane. At a City Council hearing last week, the commissioner of the Department of Buildings, Patricia Lancaster, admitted that her department had improperly approved a construction permit for the building where the crane collapsed.

In recent weeks, some elected officials and newspaper editorial boards have been putting pressure on Ms. Lancaster to go, and Mr. Bloomberg made the rare move yesterday of not immediately defending one of his agency heads against the public outcry. “I don’t think anybody should be fully satisfied with the Department of Buildings,” he told reporters. “Whether they’ve done everything they can or not is something I’m looking at.”

The billionaire businessman, who has run his administration with a management style that he developed in the private sector, is known for his fierce public support of employees, associates, and allies who come under fire for their performance in city crises. When two firefighters died in an abandoned building at ground zero last year that was later found to have not been properly inspected by the Fire Department, Mr. Bloomberg tirelessly defended his fire commissioner against calls for his resignation. He has done the same in the past for his commissioners in charge of homeless services and child welfare when their departments have faced scrutiny.

And he took a lot of heat for his refusal to condemn the head of Consolidated Edison for his leadership of the private utility company following crippling blackouts and other emergencies.

Mr. Bloomberg’s lack of defense for Ms. Lancaster could foreshadow a management shakeup at the agency, or could also be his attempt to signal to her publicly that she has to get her department in order.

Mr. Bloomberg appointed Ms. Lancaster, the first woman to head the city buildings department, in the first few months of his first term, in early 2002. His assignment to her then was to “fix the agency,” one that he recently described as “severely understaffed and deeply demoralized” when she inherited it.

Ms. Lancaster is an architect and real estate broker who served two years as deputy commissioner for design and construction in the Department of General Services for Mr. Bloomberg’s predecessor, Mayor Giuliani.

Critics of the Buildings Department have said it has been a mess since the 1990s, when the department created a “self-certification” system to streamline the permit process. Self-certification lets architects and engineers confirm on their own that some plans comply with regulations, instead of having department inspectors do it independently.

A 2003 audit by the city comptroller found that 67% of self-certified applications contained errors; a separate audit in 2004 found that inspectors rarely followed up after issuing violations at building sites, even when hazardous conditions were discovered.

The department’s inspections staff was drastically reduced in the 1990s; when Mr. Bloomberg took office, there were just 277. He has credited Ms. Lancaster for gradually bringing the numbers back up to more than 400.

And although there seems to be an increase in construction fatalities in 2008 when compared with 2007, there were 18 deaths in 2006. Mr. Bloomberg reminded buildings employees at an agency-wide meeting in February that their job is “to save lives.”

“And let me make it as clear as I can,” he said at the time. “Simply shrugging your shoulders and saying, ‘Well, after all, construction work is a dangerous occupation,’ is behavior that will not be tolerated from anyone.”

Speaking yesterday at his regular news conference, Mr. Bloomberg acknowledged that construction is hazardous and that the building boom just increases the chances that things will go wrong.