In order to calculate the second Stiefel-Whitney class of a l-connected compact Riemannian 3-symmetric space $G/K$ by BorelHirzebruch's method, we have to know the second cohomology group $H^{2}(G/K, Z_{2})\cong Hom(\pi_{2}(G/K), Z_{2})$. In this paper, we shall describe precisely the connected Lie subgroup $K$ and calculate explicitly the second homotopy group $\pi_{2}(G/K)$ in terms of the roots of $G$.

Let $A_{2n}^{(m)}$ be a generalization of a tridiagonal algebra which is defined in the introduction. In this paper it is proved that if $\varphi;A_{2n}^{(m)}\rightarrow A_{2n}^{(m)}$ is a surjective isometry, then there exists a unitary operator $U$ such that $\varphi(A)=U^{*}AU$ for all $A$ in $A_{2n}^{(m)}$ or a unitary operator $W$ such that $\varphi(A)=W^{l}AW^{*}$ for all $A$ in $A_{2n}^{(m)}$, where ${}^{t}A$ is the transpose matrix of $A$.

The main result in this note is that the connection form with respect to frames in the orthonormal frame bundle for a left invariant Riemannian metric $g$ on $SU(2)$ becomes a YangMills connection if and only if $g$ is bi-invariant.