• Problem 1 studies the  symplectic linear group (my favourite matrix Lie group).
• Problem 2 gives another characterisation of one-parameter subgroups.
• Problem 3 asks you identify the Lie bracket on $\mathfrak{gl}(n)$.
• Problem 4 is about the flows of $\varphi$-related vector fields.
• Problem 5 shows that the Lie bracket (or Lie derivative) measures the failure for two flows to commute.
• Problem 6 asks you to show that an abelian Lie group has abelian Lie algebra (you will prove the converse on the Sheet G).
• Problem 7 asks to show that an embedded Lie subgroup is automatically closed.
• Problem 8 shows that the functor $G \mapsto \mathfrak{g}$ is injective on the subcategory of simply-connected Lie groups.