What is the main characteristic of pedagogical leadership in schools?

Pedagogical leadership (also called instructional leadership) is a school-leadership approach centred on improving the quality of teaching and learning, through curriculum guidance, mentoring teachers, and monitoring classroom practice, rather than through administrative authority, facility management, or budget control.

Applying this to the question: among the four listed characteristics, only "Focus on quality of teaching" describes work aimed directly at how well teachers teach and students learn, which is the defining purpose of pedagogical leadership in a school.

Contrast by value:

• Decision-making authority describes the power to make binding organizational decisions, for example on staffing or policy, a feature of general administrative leadership, not specifically instructional work.

• Infrastructure planning concerns designing, allocating, and maintaining physical facilities and equipment, a facilities-management function.

• Focus on quality of teaching concerns strengthening how content is taught and how well students learn it, through curriculum guidance and classroom mentoring, the defining purpose of pedagogical leadership.

• Financial management concerns budgeting, resource allocation, and expenditure control, a business-administration function.

Hence, pedagogical leadership in schools is best characterised by its focus on the quality of teaching.