Question.
Under what conditions do counital coalgebras have bases consisting entirely of grouplike elements? At least in the case of finite-dimensional coalgebras, or for bialgebras (or Hopf algebras in particular), is there a simple characterization?
Definitions.
A counital (coassociative) coalgebra is a vector space $V$ over a field $K$, together with operators $\def\id{\mathrm{id}} \Delta : V \to V \otimes V \qquad\qquad \varepsilon : V \to K $ such that the following equalities hold: $\begin{gather*} (\Delta \otimes \id_V) \Delta \;=\; (\id_V \otimes \Delta) \Delta \;;\\[1ex] (\varepsilon \otimes \id_V) \Delta \;=\; \id_V \;=\; (\id_V \otimes \varepsilon) \Delta \;, \end{gather*}$ which are the coassociative property of $\Delta$ (dual to the usual associative/distributive property of multiplication) and the counital property of $\varepsilon$ (dual to the property of being a multiplicative identity).
An element $\mathbf v \in V$ is grouplike if $\Delta(\mathbf v) = \mathbf v \otimes \mathbf v$, and $\mathbf v \ne \mathbf 0$. My question is about the conditions in which there exists a basis for $V$ consisting of such elements.
Examples.
There are simple examples with and without a basis of grouplike elements. For instance, for an arbitrary field $K$ and $V$ a vector space generated by two basis vectors $ \def\r{\mathbf x} \def\i{\mathbf y} \r, \i$, if we choose $ \begin{align*} \Delta(\r) &= \r \otimes \r &\quad \varepsilon(\r) &= 1 \\ \Delta(\i) &= \i \otimes \i & \varepsilon (\i) &= 1 \end {align*}$ then $\{\r,\i\}$ itself is such a basis. In particular, we then have $\Delta(a\r + b\i) = a(\r\otimes\r) + b(\i\otimes\i) \,,$ which is a product if and only if either $a=0$ or $b=0$, so that $\{\r,\i\}$ is uniquely a basis of grouplike elements. On the other hand, a coalgebra need not have any grouplike elements at all: if we instead define $ \begin{align*} \Delta(\r) &= \r \otimes \r - \i \otimes \i &\quad \varepsilon(\r) &= 1 \\ \Delta(\i) &= \r \otimes \i + \i \otimes \r & \varepsilon (\i) &= 0 \end {align*}$ then $\Delta(a\r + b\i) = a(\r \otimes \r) + b(\r \otimes \i) + b (\i \otimes \r) - a (\i \otimes \i)\,,$ which is a product vector if and only if $a^2 = -b^2$, that is if $a = b = 0$ or $a = \pm bi$, where $i^2 = -1$. In particular, for fields such as $\mathbb R$ in which $x^2+1$ is irreducible, there are no non-trivial solutions.
Is there a characterisation of which coalgebras have such a basis? Again, if there is a simple characterization at least for the finite-dimensional case, or for bialgebras / Hopf algebras. (Of course, in the case of a bialgebra, at least the unit $\eta$ is a grouplike element.)