diff --git a/Project.toml b/Project.toml
index f76dd3f..cad0614 100644
--- a/Project.toml
+++ b/Project.toml
@@ -4,12 +4,15 @@ authors = ["Seth Axen <seth.axen@gmail.com>", "Mateusz Baran <mateuszbaran89@gma
 version = "0.1.0"
 
 [deps]
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Manifolds = "1cead3c2-87b3-11e9-0ccd-23c62b72b94e"
 ManifoldsBase = "3362f125-f0bb-47a3-aa74-596ffd7ef2fb"
 
 [compat]
+LinearAlgebra = "1.6"
 Manifolds = "0.10"
 ManifoldsBase = "0.15.16"
+julia = "1.6"
 
 [extras]
 Manifolds = "1cead3c2-87b3-11e9-0ccd-23c62b72b94e"
diff --git a/src/LieGroups.jl b/src/LieGroups.jl
index 7fea863..37d79ba 100644
--- a/src/LieGroups.jl
+++ b/src/LieGroups.jl
@@ -1,10 +1,14 @@
 module LieGroups
 
-using ManifoldsBase, Manifolds
-
-# before removing them from Manifolds.jl, let's for consistency include them here but
-# overwrite them for now. This makes moving away from that (in Manifolds.jl 0.11) here non-breaking.
-import Manifolds: apply, apply!, identity_element
+using ManifoldsBase, Manifolds, LinearAlgebra
+
+#
+#
+# = Compatibility (and a bit of type piracy for now)
+# The following imports shall be removed once Manifolds.jl does no longer define them (in 0.11)
+import Manifolds: apply, apply!, identity_element, is_identity, compose
+# Both define the following structs, so these for now lead to asking for explicit prefixes
+# Manifolds: Identity, TranslationGroup
 include("documentation_glossary.jl")
 include("interface.jl")
 
@@ -31,12 +35,11 @@ export GroupOperationAction
 
 export TranslationGroup
 
-export apply, apply!, diff_apply, diff_apply!
-export base_manifold
-export adjoint,
-    adjoint!,
-    compose,
-    compose!,
+export adjoint, adjoint!, apply, apply!
+export base_Lie_group, base_manifold
+export compose, compose!
+export diff_apply,
+    diff_apply!,
     diff_left_compose,
     diff_left_compose!,
     diff_right_compose,
@@ -47,9 +50,9 @@ export adjoint,
     inv_right_compose!
 export isapprox, is_point, is_vector
 export conjugate, conjugate!, diff_conjugate, diff_conjugate!
-export exp, exp!, log, log!
+export exp, exp!
 export identity_element, identity_element!, is_identity, inv, inv!, diff_inv, diff_inv!
-export base_Lie_group, base_manifold
-export lie_bracket, lie_bracket!
-export inv
+export lie_bracket, lie_bracket!, log, log!
+export norm
+export zero_vector
 end # module LieGroups
diff --git a/src/group_operations/addition.jl b/src/group_operations/addition.jl
index 41e92e3..cf02623 100644
--- a/src/group_operations/addition.jl
+++ b/src/group_operations/addition.jl
@@ -25,20 +25,24 @@ Base.:*(e::Identity{AdditionGroupOperation}, p) = e
 Base.:*(::Any, e::Identity{AdditionGroupOperation}) = e
 Base.:*(e::Identity{AdditionGroupOperation}, ::Identity{AdditionGroupOperation}) = e
 
-function _compose(::LieGroup{𝔽,AdditionGroupOperation}, g, h) where {𝔽}
-    return g + h
-end
-
 function _compose!(G::LieGroup{𝔽,AdditionGroupOperation}, k, g, h) where {𝔽}
     ManifoldsBase.copyto!(G, k, g + h)
     return k
 end
 
-function Base.exp(::LieGroup{𝔽,AdditionGroupOperation}, X) where {𝔽}
-    return X
-end
+"""
+"""
+Base.exp(
+    ::LieGroup{𝔽,AdditionGroupOperation}, ::Identity{AdditionGroupOperation}, X
+) where {𝔽}
 
-function ManifoldsBase.exp!(G::LieGroup{𝔽,AdditionGroupOperation}, g, X) where {𝔽}
+function ManifoldsBase.exp!(
+    G::LieGroup{𝔽,AdditionGroupOperation},
+    g,
+    ::Identity{AdditionGroupOperation},
+    X,
+    t::Number=1,
+) where {𝔽}
     return copyto!(G, g, X)
 end
 
@@ -46,9 +50,6 @@ function identity_element!(::LieGroup{𝔽,AdditionGroupOperation}, e) where {
     return fill!(e, 0)
 end
 
-function Base.inv(::LieGroup{𝔽,AdditionGroupOperation}, g) where {𝔽}
-    return -g
-end
 function inv!(G::LieGroup{𝔽,AdditionGroupOperation}, h, g) where {𝔽}
     return copyto!(G, h, -g)
 end
@@ -67,19 +68,14 @@ function is_identity(
     return true
 end
 
-function Base.log(::LieGroup{𝔽,AdditionGroupOperation}, q) where {𝔽}
-    return q
-end
-function Base.log(
-    ::LieGroup{𝔽,AdditionGroupOperation}, ::Identity{AdditionGroupOperation}
+"""
+"""
+ManifoldsBase.log(
+    G::LieGroup{𝔽,AdditionGroupOperation}, ::Identity{AdditionGroupOperation}, q
 ) where {𝔽}
-    return zero_vector(G, identity_element(G))
-end
-function ManifoldsBase.log!(G::LieGroup{𝔽,AdditionGroupOperation}, X, q) where {𝔽}
-    return copyto!(G, X, q)
-end
+
 function ManifoldsBase.log!(
-    G::LieGroup{𝔽,AdditionGroupOperation}, X, ::Identity{AdditionGroupOperation}
+    G::LieGroup{𝔽,AdditionGroupOperation}, X, ::Identity{AdditionGroupOperation}, q
 ) where {𝔽}
-    return zero_vector!(G, X, identity_element(G))
+    return copyto!(G, X, q)
 end
diff --git a/src/interface.jl b/src/interface.jl
index 05b5525..a301c54 100644
--- a/src/interface.jl
+++ b/src/interface.jl
@@ -338,7 +338,7 @@ and ``$(_tex(:exp))_{$(_math(:G))}`` denotes the  [Lie group exponential functio
 !!! note
     The Lie group exponential map is usually different from the exponential map with respect
     to the metric of the underlying Riemannian manifold ``$(_math(:M))``.
-    To access the (Riemannian) exponential map, use `exp(`[`base_manifold`](@ref)`G, g, X)`.
+    To access the (Riemannian) exponential map, use `exp(`[`base_manifold`](@ref)`(G), g, X)`.
 """
 
 @doc "$_doc_exp"
@@ -383,9 +383,15 @@ function ManifoldsBase.exp(G::LieGroup, e::Identity, X, t::Number=1)
     return h
 end
 
+# Fallback to a MethodError to avoid stack overflow
 @doc "$(_doc_exp_id)"
-ManifoldsBase.exp!(::LieGroup, h, ::Identity, X, ::Number=1)
-
+function ManifoldsBase.exp!(G::LieGroup, h, e::Identity, X, t::Number=1)
+    throw(
+        MethodError(
+            ManifoldsBase.exp!, (typeof(G), typeof(h), typeof(e), typeof(X), typeof(t))
+        ),
+    )
+end
 _doc_inv_left_compose = """
     inv_left_compose(G::LieGroup, g, h)
     inv_left_compose!(G::LieGroup, k, g, h)
@@ -494,7 +500,7 @@ This falls back to checking whether `X` is a valid point on the tangent space
 at the [`identity_element``](@ref)`]`(G)` on `G.manifold` on the [`LieGroup`](@ref)
 of `G`
 """
-function ManifoldsBase.is_point(𝔤::LieAlgebra, g; kwargs...)
+function ManifoldsBase.is_point(𝔤::LieAlgebra, X; kwargs...)
     # the manifold stored in the Fiber / Lie Algebra is the Lie Group G
     G = 𝔤.manifold
     e = identity_element(G)
@@ -638,11 +644,28 @@ function ManifoldsBase.log(G::LieGroup, e::Identity, g)
     return X
 end
 
+# explicit method error to avoid stack overflow
 @doc "$(_doc_log_id)"
-ManifoldsBase.log!(::LieGroup, X, ::Identity, g)
+function ManifoldsBase.log!(G::LieGroup, X, e::Identity, g)
+    throw(MethodError(ManifoldsBase.log!, (typeof(G), typeof(X), typeof(e), typeof(g))))
+end
+
+# Move this line already to ManifoldsBase?
+LinearAlgebra.norm(𝔤::LieAlgebra, X) = LinearAlgebra.norm(𝔤.manifold, 𝔤.point, X)
+function LinearAlgebra.norm(
+    G::LieGroup{𝔽,O}, ::Identity{O}, X
+) where {𝔽,O<:AbstractGroupOperation}
+    return LinearAlgebra.norm(G, identity_element(G), X)
+end
+LinearAlgebra.norm(G::LieGroup, g, X) = LinearAlgebra.norm(G.manifold, g, X)
 
 function ManifoldsBase.representation_size(G::LieGroup)
     return ManifoldsBase.representation_size(G.manifold)
 end
 
+Base.show(io::IO, 𝔤::LieAlgebra) = print(io, "Lie Algebra( $(𝔤.manifold) )")
 Base.show(io::IO, G::LieGroup) = print(io, "LieGroup($(G.manifold), $(G.op))")
+
+function ManifoldsBase.zero_vector(G, e::Identity)
+    return ManifoldsBase.zero_vector(G, identity_element(G))
+end
diff --git a/test/LieGroupsTestSuite.jl/LieGroupsTestSuite.jl b/test/LieGroupsTestSuite.jl/LieGroupsTestSuite.jl
index ef0d7b7..214e49e 100644
--- a/test/LieGroupsTestSuite.jl/LieGroupsTestSuite.jl
+++ b/test/LieGroupsTestSuite.jl/LieGroupsTestSuite.jl
@@ -72,9 +72,17 @@ a vector `X` from the Lie Algebra.
 * `test_log=true`: test the logarithmic map.
 """
 function test_exp_log(G::LieGroup, g, h, X; test_exp=true, test_log=true)
-    @testset "exp(G, g, X) & log(G, g, h)" begin
+    @testset "(Lie group) exp & log" begin
         𝔤 = LieAlgebra(G)
+        e = Identity(G)
         if test_exp
+            # Lie group exp
+            k1 = exp(G, e, X)
+            k2 = copy(G, g)
+            exp!(G, k2, e, X)
+            @test isapprox(G, k1, k2)
+            @test is_point(G, k1)
+            # exp
             k1 = exp(G, g, X)
             k2 = copy(G, g)
             exp!(G, k2, g, X)
@@ -82,6 +90,15 @@ function test_exp_log(G::LieGroup, g, h, X; test_exp=true, test_log=true)
             @test is_point(G, k1)
         end
         if test_log
+            # Lie group log
+            Y1 = log(G, e, g)
+            Y2 = copy(G, g)
+            log!(G, Y2, e, g)
+            @test isapprox(𝔤, Y1, Y2)
+            @test is_point(𝔤, Y1)
+            @test norm(𝔤, log(G, g, g)) ≈ 0
+            @test norm(𝔤, log(G, h, h)) ≈ 0
+            # log
             Y1 = log(G, g, h)
             Y2 = copy(G, g)
             log!(G, Y2, g, h)
@@ -91,6 +108,16 @@ function test_exp_log(G::LieGroup, g, h, X; test_exp=true, test_log=true)
             @test norm(𝔤, log(G, h, h)) ≈ 0
         end
         if test_exp && test_log
+            # Lie group exp / log
+            k1 = exp(G, e, X)
+            k2 = copy(G, g)
+            exp!(G, k2, e, X)
+            Y1 = log(G, e, k1)
+            Y2 = copy(G, g)
+            log!(G, Y2, e, k2)
+            @test isapprox(𝔤, Y1, Y2)
+            @test isapprox(𝔤, X, Y1)
+            # exp & log
             k1 = exp(G, g, X)
             k2 = copy(G, g)
             exp!(G, k2, g, X)
diff --git a/test/groups/translation.jl b/test/groups/test_translation_group.jl
similarity index 86%
rename from test/groups/translation.jl
rename to test/groups/test_translation_group.jl
index f320976..477cfbe 100644
--- a/test/groups/translation.jl
+++ b/test/groups/test_translation_group.jl
@@ -7,11 +7,11 @@ using LieGroupsTestSuite
 begin
     G = TranslationGroup(3)
     # Later maybe via auto-discover?
-    p1, p2, p3 = [1.0, 0.0, 0.0], [0.0, 3.0, 0.0], [1.1, 1.2, 3.3]
+    g1, g2, g3 = [1.0, 0.0, 0.0], [0.0, 3.0, 0.0], [1.1, 1.2, 3.3]
     X1, X2, X3 = [0.0, 1.0, 0.0], [2.0, 0.0, 0.0], [0.1, 0.2, 0.3]
     properties = Dict(
         :Name => "The Translation group",
-        :Points => [p1, p2, p3],
+        :Points => [g1, g2, g3],
         :Vectors => [X1, X2, X3],
         :Functions => [compose, exp, inv, log, show, is_identity],
     )
diff --git a/test/runtests.jl b/test/runtests.jl
index 3e6bd75..56ae699 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -6,9 +6,9 @@ using LieGroupsTestSuite
 
 function include_test(path)
     @info "Testing $path"
-    @time include(path)  # show basic timing, (this will print a newline at end)
+    @time include(path)  # show basic timing, (this prints a newline at end)
 end
 
 @testset "Lie Groups" begin
-    include_test("groups/translation.jl")
+    include_test("groups/test_translation_group.jl")
 end