Fedor Petrov's answer of my preceding question shows that my question reduces to the famous Hadamard conjecture about Hadamard matrices of order $4k$. So I decided to study this conjecture and I got the following raw idea of constructing Hadamard matrices:

If $V_i$ is a row of $H_n$ then add $V_i$ to the $i$th column.

For example: Start with $$H_4=\begin{pmatrix}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\\ 1&-1&-1&1 \end{pmatrix} $$ Let $H_{4n-4}$ be a symmetric (if it exists?) Hadamard matrix. Then by the above idea we extend this matrix to $H_{4n}$ as follows: $$H_{4n }=\begin{pmatrix}H_{4n-4}&A_{(4n-4)\times 4}\\ A^T_{4\times (4n-4)}& B_{4\times 4} \\ \end{pmatrix} $$ where $B$ is a symmetric matrix and the first row of $A$ is $(1,1,1,\dots,1)$. Thus $$AA^T=4I_{4n-4},\quad H_{4n-4}A+AB=0,\quad A^T A+B^2=4n I_4.$$

**Question:** Does this idea give an algorithm for constructing Hadamard matrices?

Any suggestion for improving this idea would be greatly appreciated.