Find all invariant subspaces, cf. Exercise 7.4.32, of a rotation in R3. Exercise 7.4.32 The subspace W

Question:

Find all invariant subspaces, cf. Exercise 7.4.32, of a rotation in R3.
Exercise 7.4.32
The subspace W of a vector space V is said to be an invariant subspace under the linear transformation L: V → V if L[w] ∈ W whenever w ∈ W. Prove that ker L and mg L are both invariant subspaces.