Lie groups and Lie algebras

What is the dimension of $SU(3)$ as a real Lie group?

Which of these matrices belong to the Lie algebra $\mathfrak{su}(2)$? (Choose all that apply.)

Which element of $SL(2, \R)$ is not of the form $e^X$ with $X \in \mathfrak{sl}(2, \R)$?

$SU(2)$ and $SO(3)$ have isomorphic Lie algebras. Which property distinguishes the two groups?

In the classification of irreducible $\mathfrak{sl}(2, \C)$-representations, why must the string $v, fv, f^2v, \dots$ eventually reach $0$?

In the representation $V_2 \otimes V_2$ of $\mathfrak{sl}(2, \C)$, what is the dimension of the eigenspace of $h$ for the eigenvalue (weight) $0$?

Decompose $V_3 \otimes V_1$ into irreducible representations $V_n$. List the values of $n$ that occur, separated by commas.

For real $n \times n$ matrices, with exp the matrix exponential, which of these Lean statements is false in general (so it could not be proved)?

Write exp_sl2(a, b, c) returning the matrix exponential of $X = \begin{pmatrix} a & b \\ c & -a \end{pmatrix}$ as a nested list, using $X^2 = \delta I$ with $\delta = a^2 + bc$: $e^X = C(\delta) I + S(\delta) X$, where $C = \cosh\sqrt\delta$, $S = \sinh\sqrt\delta/\sqrt\delta$ if $\delta > 0$; $C = \cos\sqrt{-\delta}$, $S = \sin\sqrt{-\delta}/\sqrt{-\delta}$ if $\delta < 0$; and $C = S = 1$ if $\delta = 0$.