Test a few more defaults on the interface,

src/interface.jl

@@ -143,6 +143,11 @@ Return the manifold stored within the [`LieGroup`](@ref) `G`.
 """
 Manifolds.base_manifold(G::LieGroup) = G.manifold
 
+function ManifoldsBase.check_point(
+    G::LieGroup{𝔽,O}, g; kwargs...
+) where {𝔽,O<:AbstractGroupOperation}
+    return ManifoldsBase.check_point(G.manifold, g; kwargs...)
+end
 function ManifoldsBase.check_point(
     G::LieGroup{𝔽,O}, e::Identity{O}; kwargs...
 ) where {𝔽,O<:AbstractGroupOperation}
@@ -532,8 +537,7 @@ This falls back to checking whether `g` is a valid point on `G.manifold`,
 unless `g` is an [`Identity`](@ref). Then, it is checked whether it is the
 idenity element corresponding to `G`.
 """
-ManifoldsBase.is_point(G::LieGroup, g; kwargs...) =
-    ManifoldsBase.is_point(G.manifold, g; kwargs...)
+ManifoldsBase.is_point(G::LieGroup, g; kwargs...)
 
 _doc_is_vector = """
     is_vector(G::LieGroup, X; kwargs...)

test/test_interface.jl

@@ -16,4 +16,14 @@ begin
     @test repr(𝔤) == rs2
     @test is_identity(G, Identity(op))
     @test !is_identity(G, Identity(op2))
+    e = Identity(op)
+    @test compose(G, e, e) == e
+    @test is_point(G, e)
+    @test !is_point(G, Identity(op2))
+    @test_throws DomainError is_point(G, Identity(op2); error=:error)
+    # Exp log Method Error fallbacks that avoid the stack overflow
+    g = :none
+    X = :nonetoo
+    @test_throws MethodError exp!(G, g, e, X)
+    @test_throws MethodError log!(G, X, e, g)
 end