Even before she faces a parliament vote to become Japan’s first woman Prime Minister, Sanae Takaichi has set a record as the first woman to lead Japan’s dominant Liberal Democratic Party (LDP). After defeating her more centrist rival Shinjiro Koizumi in party elections on Friday, Ms. Takaichi, a 64-year-old protégée of former Prime Minister Shinzo Abe, is seen as a socially conservative right-winger. If she wins the parliamentary vote, as expected, she faces more difficult challenges than breaking the decades-old glass ceiling. To begin with, Ms. Takaichi is the latest in a series of Japanese Prime Ministers ousted from office, even as the LDP and its partner Komeito have lost majorities in both houses, and will need to lead a fractious coalition by reaching out to the opposition. Her pr

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