This study investigates the interplay between alternagnetism and weak ferromagnetism in two-dimensional RuF₄. Using density functional theory and symmetry analysis, the researchers show that RuF₄ monolayer is a two-dimensional d-wave alternagnet. Spin-orbit coupling leads to significant spin splitting of electronic bands at the Γ point, up to ~100 meV, and induces weak ferromagnetism due to non-trivial spin-momentum locking, which tilts the Ru magnetic moments. The net magnetic moment scales linearly with the spin-orbit coupling strength. An effective spin Hamiltonian is derived to capture the spin-splitting and spin-momentum locking of the electronic bands. The study reveals that alternagnetism and spin-orbit coupling-induced spin splitting are closely related, with the latter affecting the former. The results highlight the non-trivial spin-momentum locking and weak ferromagnetism in two-dimensional alternagnets, which are important for future applications in spintronics. The structure and crystal symmetries of RuF₄ monolayer are analyzed, showing that it belongs to the p2₁/b11 layer group. The magnetic properties of RuF₄ are discussed, showing that it has a compensated net magnetic moment due to opposite magnetic moments on the two Ru sublattices. The spin splitting of the electronic bands is analyzed, showing that the top valence and bottom conduction bands have the largest spin splitting. The inclusion of spin-orbit coupling turns RuF₄ into a weak ferromagnet, with the magnetic moment pointing in the +x direction. The study also shows that the spin splitting of the electronic bands is significantly affected by spin-orbit coupling, with the spin splitting reaching its maximum at the Γ point. The results demonstrate the importance of spin-orbit coupling in the magnetic properties of RuF₄ and its potential for applications in spintronics.This study investigates the interplay between alternagnetism and weak ferromagnetism in two-dimensional RuF₄. Using density functional theory and symmetry analysis, the researchers show that RuF₄ monolayer is a two-dimensional d-wave alternagnet. Spin-orbit coupling leads to significant spin splitting of electronic bands at the Γ point, up to ~100 meV, and induces weak ferromagnetism due to non-trivial spin-momentum locking, which tilts the Ru magnetic moments. The net magnetic moment scales linearly with the spin-orbit coupling strength. An effective spin Hamiltonian is derived to capture the spin-splitting and spin-momentum locking of the electronic bands. The study reveals that alternagnetism and spin-orbit coupling-induced spin splitting are closely related, with the latter affecting the former. The results highlight the non-trivial spin-momentum locking and weak ferromagnetism in two-dimensional alternagnets, which are important for future applications in spintronics. The structure and crystal symmetries of RuF₄ monolayer are analyzed, showing that it belongs to the p2₁/b11 layer group. The magnetic properties of RuF₄ are discussed, showing that it has a compensated net magnetic moment due to opposite magnetic moments on the two Ru sublattices. The spin splitting of the electronic bands is analyzed, showing that the top valence and bottom conduction bands have the largest spin splitting. The inclusion of spin-orbit coupling turns RuF₄ into a weak ferromagnet, with the magnetic moment pointing in the +x direction. The study also shows that the spin splitting of the electronic bands is significantly affected by spin-orbit coupling, with the spin splitting reaching its maximum at the Γ point. The results demonstrate the importance of spin-orbit coupling in the magnetic properties of RuF₄ and its potential for applications in spintronics.