Let S be a nonassociative k-algebra. By using the Lie triple system, we study the subspace I2（S） of the Steinberg Lie algebra st2（S） and give a necessary and sufficient condition for I2（S） = 0.
Seventy years ago, Myers and Steenrod showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. In 2007, Bagaev and Zhukova proved the same result for a Riemannian orbifold. In this paper, the authors first show that the isometry group of a Riemannian manifold M with boundary has dimension at most 1/2 dim M（dim M - 1）. Then such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension are completely classified.
