设 $A=\left(\alpha_1, \alpha_2, \alpha_3, \alpha_4\right)$ 为 4 阶正交矩阵，若矩阵
$B=\left(\begin{array}{l}\alpha_1^T \\ \alpha_2^T \\ \alpha_3^T\end{array}\right), \beta=\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right), k$ 为任意常数，
则线性方程组 $B x=\beta$ 的通解为 ( $\quad$ ）
A. $\alpha_2+\alpha_3+\alpha_4+k \alpha_1$
B. $\alpha_1+\alpha_3+\alpha_4+k \alpha_2$
C. $\alpha_1+\alpha_2+\alpha_4+k \alpha_3$
D. $\alpha_1+\alpha_2+\alpha_3+k \alpha_4$