Time complexity for multiplying two lower triangular matrices
With the expert hints of Mr Emil, I could find a reduction of general matrix multiplication to triangular matrix multiplication. If we wish to multiply two $n \times n$ matrices $A$ and $B$, I can embed $A$ as $M_{32}^{th}$ block of a $3n \times 3n$ matrix $M$ with rest of the blocks all zero matrices. Similarly, I can embed $B$ as $N_{21}^{th}$ block of $3n \times 3n$ matrix $N$ with rest of the blocks all zero. Both M and N are lower triangular. An asymptotically faster algorithm for $M.N$ would then yield faster algorithm for $A.B$ too. 

$$
\begin{bmatrix}
0 &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 0\\
0 &amp; A &amp; 0
\end{bmatrix}
\begin{bmatrix}
0 &amp; 0 &amp; 0\\
B &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 0
\end{bmatrix}
=
\begin{bmatrix}
0 &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 0\\
AB &amp; 0 &amp; 0
\end{bmatrix}
$$

It's amazing how this argument does not manifest when trying to embed $A$ and $B$ in $2n \times 2n$ matrices!