Riesz bases consisting of root functions of 1D Dirac operators

For one-dimensional Dirac operators $$Ly=i\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\frac{dy}{dx}+vy,v=\left(\begin{array}{cc}0& P\\ Q& 0\end{array}\right),y=\left(\begin{array}{c}{y}_1\\ y_2\end{array}\right),$$ subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in $L^2([0,\pi ],{\mathbb{C}}^2)\mathrm{.}$ In particular, if the potential matrix $v$ is skew-symmetric (i.e., $\bar{Q}=-P$), or more generally if $\bar{Q}=tP$ for some real $t \neq 0,$ then there exists a Riesz basis that consists of root functions of the operator $L\mathrm{.}$