Normality Relation is not Transitive/Proof 1

Proof
Proof by Counterexample:

Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as:

From Normality of Subgroups of Alternating Group on 4 Letters:
 * $K := \set {e, t, u, v}$ is a normal subgroup of $A_4$
 * $T := \set {e, t}$ is not a normal subgroup of $A_4$.

But by Subgroup of Abelian Group is Normal:
 * $T$ is a normal subgroup of $K$.

Thus we have:
 * $T \lhd K$, $K \lhd A_4$

but:
 * $T \not \lhd A_4$