diff --git a/experimental/LieAlgebras/src/LieAlgebras.jl b/experimental/LieAlgebras/src/LieAlgebras.jl
index 9b264bf3ff43..d6cd094e36c6 100644
--- a/experimental/LieAlgebras/src/LieAlgebras.jl
+++ b/experimental/LieAlgebras/src/LieAlgebras.jl
@@ -84,6 +84,7 @@ import ..Oscar:
   tensor_product,
   weyl_vector,
   word,
+  zero!,
   zero_map,
   ⊕,
   ⊗
diff --git a/experimental/LieAlgebras/src/RootSystem.jl b/experimental/LieAlgebras/src/RootSystem.jl
index 902e6773741a..88404e9b235d 100644
--- a/experimental/LieAlgebras/src/RootSystem.jl
+++ b/experimental/LieAlgebras/src/RootSystem.jl
@@ -524,6 +524,14 @@ function RootSpaceElem(w::WeightLatticeElem)
   return RootSpaceElem(root_system(w), w)
 end
 
+function zero(::Type{RootSpaceElem}, R::RootSystem)
+  return RootSpaceElem(R, zero_matrix(QQ, 1, rank(R)))
+end
+
+function zero(r::RootSpaceElem)
+  return zero(RootSpaceElem, root_system(r))
+end
+
 function Base.:*(q::RationalUnion, r::RootSpaceElem)
   return RootSpaceElem(root_system(r), q * r.vec)
 end
@@ -544,6 +552,29 @@ function Base.:-(r::RootSpaceElem)
   return RootSpaceElem(root_system(r), -r.vec)
 end
 
+function zero!(r::RootSpaceElem)
+  r.vec = zero!(r.vec)
+  return r
+end
+
+function add!(rr::RootSpaceElem, r1::RootSpaceElem, r2::RootSpaceElem)
+  @req root_system(rr) === root_system(r1) === root_system(r2) "parent root system mismatch"
+  rr.vec = add!(rr.vec, r1.vec, r2.vec)
+  return rr
+end
+
+function neg!(rr::RootSpaceElem, r::RootSpaceElem)
+  @req root_system(rr) === root_system(r) "parent root system mismatch"
+  rr.vec = neg!(rr.vec, r.vec)
+  return rr
+end
+
+function sub!(rr::RootSpaceElem, r1::RootSpaceElem, r2::RootSpaceElem)
+  @req root_system(rr) === root_system(r1) === root_system(r2) "parent root system mismatch"
+  rr.vec = sub!(rr.vec, r1.vec, r2.vec)
+  return rr
+end
+
 function Base.:(==)(r::RootSpaceElem, r2::RootSpaceElem)
   return r.root_system === r2.root_system && r.vec == r2.vec
 end
@@ -682,6 +713,14 @@ function DualRootSpaceElem(root_system::RootSystem, vec::Vector{<:RationalUnion}
   return DualRootSpaceElem(root_system, matrix(QQ, 1, length(vec), vec))
 end
 
+function zero(::Type{DualRootSpaceElem}, R::RootSystem)
+  return DualRootSpaceElem(R, zero_matrix(QQ, 1, rank(R)))
+end
+
+function zero(r::DualRootSpaceElem)
+  return zero(DualRootSpaceElem, root_system(r))
+end
+
 function Base.:*(q::RationalUnion, r::DualRootSpaceElem)
   return DualRootSpaceElem(root_system(r), q * r.vec)
 end
@@ -702,6 +741,29 @@ function Base.:-(r::DualRootSpaceElem)
   return DualRootSpaceElem(root_system(r), -r.vec)
 end
 
+function zero!(r::DualRootSpaceElem)
+  r.vec = zero!(r.vec)
+  return r
+end
+
+function add!(rr::DualRootSpaceElem, r1::DualRootSpaceElem, r2::DualRootSpaceElem)
+  @req root_system(rr) === root_system(r1) === root_system(r2) "parent root system mismatch"
+  rr.vec = add!(rr.vec, r1.vec, r2.vec)
+  return rr
+end
+
+function neg!(rr::DualRootSpaceElem, r::DualRootSpaceElem)
+  @req root_system(rr) === root_system(r) "parent root system mismatch"
+  rr.vec = neg!(rr.vec, r.vec)
+  return rr
+end
+
+function sub!(rr::DualRootSpaceElem, r1::DualRootSpaceElem, r2::DualRootSpaceElem)
+  @req root_system(rr) === root_system(r1) === root_system(r2) "parent root system mismatch"
+  rr.vec = sub!(rr.vec, r1.vec, r2.vec)
+  return rr
+end
+
 function Base.:(==)(r::DualRootSpaceElem, r2::DualRootSpaceElem)
   return r.root_system === r2.root_system && r.vec == r2.vec
 end
@@ -839,18 +901,26 @@ function WeightLatticeElem(r::RootSpaceElem)
   return WeightLatticeElem(root_system(r), r)
 end
 
+function zero(::Type{WeightLatticeElem}, R::RootSystem)
+  return WeightLatticeElem(R, zero_matrix(ZZ, rank(R), 1))
+end
+
+function zero(r::WeightLatticeElem)
+  return zero(WeightLatticeElem, root_system(r))
+end
+
 function Base.:*(n::IntegerUnion, w::WeightLatticeElem)
   return WeightLatticeElem(root_system(w), n * w.vec)
 end
 
 function Base.:+(w::WeightLatticeElem, w2::WeightLatticeElem)
-  @req root_system(w) === root_system(w2) "parent weight lattics mismatch"
+  @req root_system(w) === root_system(w2) "parent root system mismatch"
 
   return WeightLatticeElem(root_system(w), w.vec + w2.vec)
 end
 
 function Base.:-(w::WeightLatticeElem, w2::WeightLatticeElem)
-  @req root_system(w) === root_system(w2) "parent weight lattics mismatch"
+  @req root_system(w) === root_system(w2) "parent root system mismatch"
 
   return WeightLatticeElem(root_system(w), w.vec - w2.vec)
 end
@@ -859,25 +929,29 @@ function Base.:-(w::WeightLatticeElem)
   return WeightLatticeElem(root_system(w), -w.vec)
 end
 
+function zero!(w::WeightLatticeElem)
+  w.vec = zero!(w.vec)
+  return w
+end
+
 function add!(wr::WeightLatticeElem, w1::WeightLatticeElem, w2::WeightLatticeElem)
-  add!(wr.vec, w1.vec, w2.vec)
+  @req root_system(wr) === root_system(w1) === root_system(w2) "parent root system mismatch"
+  wr.vec = add!(wr.vec, w1.vec, w2.vec)
   return wr
 end
 
-add!(w1::WeightLatticeElem, w2::WeightLatticeElem) = add!(w1, w1, w2)
-
-function neg!(w::WeightLatticeElem)
-  neg!(w.vec)
-  return w
+function neg!(wr::WeightLatticeElem, w::WeightLatticeElem)
+  @req root_system(wr) === root_system(w) "parent root system mismatch"
+  wr.vec = neg!(wr.vec, w.vec)
+  return wr
 end
 
 function sub!(wr::WeightLatticeElem, w1::WeightLatticeElem, w2::WeightLatticeElem)
-  sub!(wr.vec, w1.vec, w2.vec)
+  @req root_system(wr) === root_system(w1) === root_system(w2) "parent root system mismatch"
+  wr.vec = sub!(wr.vec, w1.vec, w2.vec)
   return wr
 end
 
-sub!(w1::WeightLatticeElem, w2::WeightLatticeElem) = sub!(w1, w1, w2)
-
 function Base.:(==)(w::WeightLatticeElem, w2::WeightLatticeElem)
   return w.root_system === w2.root_system && w.vec == w2.vec
 end
@@ -983,7 +1057,7 @@ function conjugate_dominant_weight_with_elem!(w::WeightLatticeElem)
 end
 
 function dot(w1::WeightLatticeElem, w2::WeightLatticeElem)
-  @req root_system(w1) === root_system(w2) "parent weight lattices mismatch"
+  @req root_system(w1) === root_system(w2) "parent root system mismatch"
   R = root_system(w1)
 
   return dot(
diff --git a/experimental/LieAlgebras/test/RootSystem-test.jl b/experimental/LieAlgebras/test/RootSystem-test.jl
index 278000c23b73..b161be1a300e 100644
--- a/experimental/LieAlgebras/test/RootSystem-test.jl
+++ b/experimental/LieAlgebras/test/RootSystem-test.jl
@@ -132,6 +132,79 @@
         )
       end
 
+      @testset "Mutating arithmetics for $T" for T in (
+        RootSpaceElem, DualRootSpaceElem, WeightLatticeElem
+      )
+        rk = rank(R)
+
+        x1 = T(R, rand(-10:10, rk))
+        x2 = T(R, rand(-10:10, rk))
+        x3 = T(R, rand(-10:10, rk))
+        r1c = deepcopy(x1)
+        r2c = deepcopy(x2)
+        r3c = deepcopy(x3)
+
+        for (f, f!) in ((zero, zero!),)
+          x1 = f!(x1)
+          @test x1 == f(r1c)
+          x1 = deepcopy(r1c)
+        end
+
+        for (f, f!) in ((-, neg!),)
+          x1 = f!(x1, x2)
+          @test x1 == f(r2c)
+          @test x2 == r2c
+          x1 = deepcopy(r1c)
+          x2 = deepcopy(r2c)
+
+          x1 = f!(x1)
+          @test x1 == f(r1c)
+          x1 = deepcopy(r1c)
+        end
+
+        for (f, f!) in ((+, add!), (-, sub!))
+          x1 = f!(x1, x2, x3)
+          @test x1 == f(r2c, r3c)
+          @test x2 == r2c
+          @test x3 == r3c
+          x1 = deepcopy(r1c)
+          x2 = deepcopy(r2c)
+          x3 = deepcopy(r3c)
+
+          x1 = f!(x1, x1, x2)
+          @test x1 == f(r1c, r2c)
+          @test x2 == r2c
+          x1 = deepcopy(r1c)
+          x2 = deepcopy(r2c)
+
+          x1 = f!(x1, x2, x1)
+          @test x1 == f(r2c, r1c)
+          @test x2 == r2c
+          x1 = deepcopy(r1c)
+          x2 = deepcopy(r2c)
+
+          x1 = f!(x1, x2, x2)
+          @test x1 == f(r2c, r2c)
+          @test x2 == r2c
+          x1 = deepcopy(r1c)
+          x2 = deepcopy(r2c)
+
+          x1 = f!(x1, x1, x1)
+          @test x1 == f(r1c, r1c)
+          x1 = deepcopy(r1c)
+
+          x1 = f!(x1, x2)
+          @test x1 == f(r1c, r2c)
+          @test x2 == r2c
+          x1 = deepcopy(r1c)
+          x2 = deepcopy(r2c)
+
+          x1 = f!(x1, x1)
+          @test x1 == f(r1c, r1c)
+          x1 = deepcopy(r1c)
+        end
+      end
+
       @testset "Serialization" begin
         mktempdir() do path
           test_save_load_roundtrip(path, R) do loaded