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 get the following raw idea in constructing Hadamard matrices:

If $V_i$ is a row of $H_n$ then add $V_i$ to the $i$th column. for example: $$H_4=\begin{pmatrix}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\\ 1&-1&-1&1 \end{pmatrix} $$ Let $H_{4n-4}$ be a symmetric (if exist?) Hadamard matrix. then by above idea we extend this matrix to $H_{4n}$ as follow $$H_{4n }=\begin{pmatrix}H_{4n-4}&A_{(4n-4)\times 4}\\ A^t_{4\times (4n-4)}& B_{4\times 4} \\ \end{pmatrix} $$ where $B$ is a symmetric matrix and firs row of $A$ is $(1,1,1,\cdots,1)$. 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 gives an algorithm for constructing Hadamard matrices?

Any suggestion for improving this idea would be greatly appreciate.

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.