Let \(\mathfrak {L}_{\mathfrak {sl}_2}\) be the semi-direct product of the Witt algebra and the Lie algebra \(\mathfrak {sl}_2\) . In this paper, we determine the second cohomology of \(\mathfrak {L}_{\mathfrak {sl}_2}\) with trivial coefficients and determine its universal central extension. This yields two 2-cocycles. We then proceed to we consider the universal central extension of the Lie algebra \(\mathfrak {L}_{\mathfrak {sl}_2}\) in the category of Leibniz algebras. We then calculate the first cohomology group of \(\mathfrak {L}_{\mathfrak {sl}_2}\) in the adjoint module to classify all derivations of \(\mathfrak {L}_{\mathfrak {sl}_2}\) . We also study the 2-local derivations and biderivations of \(\mathfrak {L}_{\mathfrak {sl}_2}\) . Finally, based on the result of biderivations, we determine the linear commuting maps on \(\mathfrak {L}_{\mathfrak {sl}_2}\) .
