Nevertheless, Rodriguez received one.

It is not clear exactly how Rodriguez made his initial case to Smith in 2007 so that he could use testosterone. According to one baseball official, who declined to be named because he did not want to be quoted publicly discussing a major league player, Rodriguez used a doctor to provide medical information to Smith about his condition, identified in the book as hypogonadism, which occurs when the body does not produce enough natural testosterone.

According to the book, Major League Baseball’s top drug-testing official, Rob Manfred, testified about the 2007 exemption in connection with the appeal that Rodriguez filed last fall after he was hit with an initial 211-game suspension because of his involvement in the Biogenesis drug-distribution scandal in South Florida.

In that testimony, the book said, Manfred said that testosterone was “the mother of all anabolics,” a powerful performance enhancer.

Exemptions to use such substances in baseball, Manfred went on to say in the testimony, were “very rare.” He said that was because “some people who have been involved in this field feel that with a young male, healthy young male, the most likely cause of low testosterone requiring this type of therapy would be prior steroid abuse.”

Yet while Rodriguez’s request to use testosterone in 2007 may have been a tipoff that he had previously used steroids as a major leaguer, it apparently did not deter Smith, who went forward with approving it.

“If you have a player who has low testosterone because he has used for years and is no longer using, do you treat him or not?” the baseball official said Wednesday in defending Smith’s decision back in 2007. Of Rodriguez, he said: “Everyone knows now that he’s a steroid user, but at the time we didn’t have anything to go on.”

In 2012, Smith was dismissed as the doctor overseeing the drug program because the players union felt that he had made it too difficult for players to receive exemptions, according to two baseball officials. However, Smith still oversees baseball’s minor league testing program, which is outside the union’s domain.