Affine extensions of ${\mathbb{Z}}_2^2$-graded $osp(1\mathrm{\mid }2)$ and Virasoro algebra

It is known that there are two inequivalent ${\mathbb{Z}}_2^2$-graded $osp(1\mathrm{\mid }2)$ Lie superalgebras. Their affine extensions are investigated and it is shown that one of them admits two central elements, one is non-graded and the other is $(1,1)$-graded. The affine ${\mathbb{Z}}_2^2$-$osp(1\mathrm{\mid }2)$ algebras are used by the Sugawara construction to study possible ${\mathbb{Z}}_2^2$-graded extensions of the Virasoro algebra. We obtain a ${\mathbb{Z}}_2^2$-graded Virasoro algebra with a non-trivially graded central element. Throughout the investigation, invariant bilinear forms on ${\mathbb{Z}}_2^2$-graded superalgebras play a crucial role, so a theory of invariant bilinear forms is also developed.