Boeing's CEO defended the aerospace company's safety inspection system in Senate testimony Tuesday, despite two crashes that have grounded its 737 Max jetliner.

He also disclosed he was notified before the second crash of a test pilot's signal of "egregious" problems with the Max's flight control system, now believed to be at fault in the plane's two crashes.

Overall, however, CEO Dennis Muilenburg took an apologetic stance before the Senate Commerce, Science, and Transportation Committee. The hearing was attended by family members of the 346 people who lost their lives in the crashes. They stood at one point to show large photos of their deceased relatives.

Not only will the 737 Max not fly again until all agree it is safe, Muilenburg said, but Boeing also is reviewing all its safety policies.

"On behalf of myself and the Boeing Company, we are deeply and truly sorry. As a husband and father myself, I am heartbroken by your losses. I think about you and your loved ones every day, and the entire Boeing team does as well," he said.

But when pressed on specifics by senators, Muilenburg took a harder line.

Even though he knew about the 2016 electronic messages from the test pilot, he denied knowing until recently the specifics they contained. The pilot talked about unknowingly lying to regulators. The company turned the messages over to the Justice Department in February, but didn't share it with the Federal Aviation Administration and the Senate committee until recently.

Safety is ingrained in the culture at Boeing, the CEO repeatedly said. But Sen. Ted Cruz, R-Texas, pressed him about the 2016 message exchange.

"This exchange is stunning," Cruz said, adding that he was aghast that the exchange wasn't shared earlier.

Muilenburg said he was made aware of the messages earlier this year, but didn't learn the details until recently. He said the pilot has left Boeing and the only communication with him has been through his lawyer. "I fully support diving deep into this and understanding what he meant," Muilenberg said.

He also said he was open to improvements in a system that designates Boeing employees to make inspections on behalf of the FAA. The Senate panel, as well as a House committee where Muilenburg is scheduled to appear Wednesday, has questioned whether the system's self-regulation is too cozy.

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"We believe that the delegated authority process that we have today has contributed to improved safety over the past two decades," Muilenburg said. "We are open to improving it. But the idea we can tap the deep technical expertise of our companies across the aerospace industry is a valuable part of the certification process.

"But the FAA is the certification authority and should be."

After the hearing, Muilenburg met privately with relatives of people who died in the two Max crashes.

"He expressed contrition. We expressed frustration," said Michael Stumo of Sheffield, Mass., who lost his daughter Samya Rose Stumo in the Ethiopian Airlines crash of a 737 Max jet.

The relatives were miffed that Muilenburg had apologized to them while facing the committee, not turning around to face them in the audience, Stumo said.

Surviving family members want to see action before the 737 Max can fly again, he said. Muilenburg took under advisement their request to see a public demonstration of the improvements to the automated flight control system, Stumo said, instead of just having it tested by pilots and approved by the FAA.

Stumo thinks the visit made an impression on Muilenburg.

"He said his team felt the emotion in the room and promised to do better," he said.

Correction: A previous version of this story gave the incorrect day for Muilenburg's Senate testimony. He testified on Tuesday.