Using induction to prove an orthogonal matrix

Question

If $A$ is a function of $(x_1,x_2,\cdot\cdot\cdot,x_{2^{k_0}})$, denote as $A(x_1,x_2,\cdot\cdot\cdot,x_{2^{k_0}})$, and $A$ is an orthogonal matrix. Let $B=A(x_{n_0+1},x_{n_0+2},\cdot\cdot\cdot,x_{2n_0})$ and $A=A(x_{1},x_{2},\cdot\cdot\cdot,x_{n_0})$, How to use induction prove that $B^TA=A^TB$ and $$\begin{bmatrix}A & B \\B^T & -A^T\end{bmatrix}$$