 Solution

An orthogonal matrix has orthonormal columns. The first column of $L$ has squared norm $1+c^2$, so $c=0$. Orthogonality of the two columns then gives $-b+cd=-b=0$, hence $b=0$. The second column must have squared norm $d^2=1$. ThereforeBoth resulting matrices satisfy $L^{T}L=I$ directly.