Matrix groups and continuous symmetry

Which of these real $3 \times 3$ matrices lie in $SO(3)$? (Choose all that apply.)

A rotation $A \in SO(3)$ has $\operatorname{tr} A = 0$. Through what angle $\theta \in [0°, 180°]$ does it rotate, in degrees?

What is the dimension of the rotation group $SO(5)$ (equivalently of its Lie algebra $\mathfrak{so}(5)$)?

Which of these matrix groups are compact? (Choose all that apply.)

The proof of Euler's theorem shows $\det(A - I) = -\det(A - I)$ for $A \in SO(3)$. Which step uses that the dimension is odd?

The unit quaternion $q_\theta = \cos\frac\theta2 + \sin\frac\theta2\,k$ induces the rotation $\rho(q_\theta)$ by $\theta$ about the $z$-axis. As $\theta$ increases from $0°$ to $360°$, the rotation returns to the identity. What is the real part of $q_\theta$ at $\theta = 360°$?

The Lean theorem exp_skew_orthogonal concludes (NormedSpace.exp X)ᵀ * NormedSpace.exp X = 1 for a real square matrix X. Which hypothesis makes it true?

For which pairs is $\exp(X + Y) = \exp X \exp Y$? Here $N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, $N' = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, $J = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, $E = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$. (Choose all that apply.)

Implement Hamilton's product qmul(p, q) for quaternions given as tuples (a, b, c, d) meaning $a + bi + cj + dk$, and rotate(q, v) returning the vector part of $q\,v\,\bar q$ for a unit quaternion q and a vector v = (x, y, z) (identified with the pure quaternion $xi + yj + zk$).