f4 — compact exceptional Lie algebra

The compact real Lie algebra of type F4 is the compact real form of $\mathfrak{f}_{4}(\mathbb C)$ . Equivalently it is a real Lie algebra whose complexification is that simple complex algebra and whose Killing form is negative definite. It is unique up to real Lie algebra isomorphism.

Construction and dimensions

In a compatibly normalized root basis, a concrete model is the real span of

i h_j,\qquad e_\alpha-e_{-\alpha},\qquad i(e_\alpha+e_{-\alpha})\quad(\alpha>0).

Its real dimension is 52; its complexification has complex rank 4. This compact real object is distinct from the split real form and from the underlying real algebra of the complex form.

References

Knapp, Lie Groups Beyond an Introduction, digital second edition, §VI.1, Corollary 6.10 and Theorem 6.11, pp. 352–354; Appendix C.2, pp. 686–692.