If $L,A \in m(n,n)$ on $R$, when exists L for which $(L^{T}AL)$ is a diagonal block matrix?

Question

If $L,A \in m(n,n)$ on $R$, when exists L for which $(L^{T}AL)$ is a diagonal block matrix? Of course if $A$ is symmetrycal then $L$ exists and is a matrix of its eigenvectors, but I don't want to consider this case.