How can boron form three bonds or four bonds in compounds?

Boron forms three or four bonds through the process of hybridization, which allows it to accommodate different coordination numbers despite its electron configuration.1

Three-Bonded Boron

To form three bonds, boron promotes one of its 2s electrons into a 2p orbital, resulting in an outer electron configuration of $2s^{1}2p^2$. These orbitals mix to create three $s{p}^2$ hybrid orbitals. This configuration results in planar molecules, such as boron trifluoride, where the boron atom is three-coordinated and possesses an incomplete octet.123

Four-Bonded Boron

Boron achieves four-coordinated structures by utilizing $s{p}^3$ hybridization. This occurs when a three-coordinated boron compound (a Lewis acid) accepts an unshared pair of electrons from a donor atom or molecule to form an adduct. In these complexes, the four groups are tetrahedrally disposed around the boron atom. Additionally, boron can form four bonds in structures like metal borohydrides, where the $s{p}^3$ configuration is also employed.12

Beyond these standard covalent bonds, boron also participates in unique three-centre, two-electron bonds, which are essential for the structure of boranes and other complex polyhedral clusters.24